Introduction to Operations Research

9th Edition Hillier Solutions Manual

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 CHAPTER 10: DYNAMIC PROGRAMMING

10.2-1.
(a) The nodes of the network can be divided into "layers" such that the nodes in the 8th
layer are accessible from the origin only through the nodes in the Ð8  "Ñst layer. These
layers define the stages of the problem, which can be labeled as 8 œ "ß #ß $ß %. The nodes
constitute the states.
Let W8 denote the set of the nodes in the 8th layer of the network, i.e., W" œ ÖS×, W# œ
ÖEß Fß G×, W$ œ ÖHß I× and W% œ ÖX ×. The decision variable B8 is the immediate
destination at stage 8. Then the problem can be formulated as follows:
08‡ Ð=Ñ œ ‡
min Ò-=B8  08" ÐB8 ÑÓ ´ min 08 Ð=ß B8 Ñ for = − W8 and 8 œ "ß #ß $
B8 −W8" B8 −W8"

0%‡ ÐX Ñ œ !
(b) The shortest path is S  F  H  X .

(c) Number of stages: 3

=$ 0$‡ Ð=Ñ B‡$
H 6 X
I ( X
=# 0# Ð=ß HÑ 0# Ð=ß IÑ 0#‡ Ð=Ñ B‡#
E ""  "" H
F "$ "& "$ H
G  "$ "$ I
=" 0" Ð=ß EÑ 0" Ð=ß FÑ 0" Ð=ß GÑ 0"‡ Ð=Ñ B‡"
S #! "* #! "* F
Optimal Solution: B‡" œ F , B‡# œ H and B‡$ œ H.

10-1
(d) Shortest-Path Algorithm:
Solved nodes Closest 8th Distance to
directly connected connected total nearest 8th nearest Last
8 to unsolved nodes unsolved node distance node node connection
1 S F ' F ' SF
# S G ( G ( SG
F H '  ( œ "$
$ S E * E * SE
F H '  ( œ "$
G I (  ' œ "$
% E H *  & œ "% H "$ FH
F H '  ( œ "$
G I (  ' œ "$ I GI
& H X "$  ' œ "* X "* HX
I X "$  ( œ #!

The shortest-path algorithm required ) additions and ' comparisons whereas dynamic
programming required ( additions and $ comparisons. Hence, the latter seems to be more
efficient for shortest-path problems with "layered" network graphs.
10.2-2.
(a)

The optimal routes are S  E  J  X and S  G  L  X , the associated sales

income is "&(. The route S  E  J  X corresponds to assigning ", #, and $
salespeople to regions ", #, and $ respectively. The route S  G  L  X corresponds
to assigning $, #, and " salespeople to regions ", #, and $ respectively.

10-2
(b) The regions are the stages and the number of salespeople remaining to be allocated at
stage 8 are possible states at stage 8, say =8 . Let B8 be the number of salespeople
assigned to region 8 and -8 ÐB8 Ñ be the increase in sales in region 8 if B8 salespeople are
assigned to it. Number of stages: 3.
=$ 0$‡ Ð=$ Ñ B‡$ 0# Ð=# ß B# Ñ
" $# " =# " # $ % 0#‡ Ð=# Ñ B‡#
# %' # # &'    &' 1
$ (! $ $ (! (*   (* #
% )% % % *% *$ *&  *& $
& "!) ""( "!* ""! ""( #
0" Ð=" ß B" Ñ
=" " # $ % 0"‡ Ð=" Ñ B‡"
' "&( "%* "&( "&& "&( "ß $
The optimal solutions are ÐB‡" œ ", B‡# œ #, B‡$ œ $Ñ and ÐB‡" œ $, B‡# œ #, B‡$ œ "Ñ.
10.2-3.
(a) The five stages of the problem correspond to the five columns of the network graph.
The states are the nodes of the graph. Given the activity times >34 , the problem can be
formulated as follows:
08‡ Ð=Ñ œ max Ò>=B8  08"
‡
ÐB8 ÑÓ
B8

0'‡ Ð*Ñ œ !
(b) The critical paths are " Ä # Ä % Ä ( Ä * and " Ä # Ä & Ä ( Ä *.

10-3
(c) Interactive Deterministic Dynamic Programming Algorithm: Number of stages: 4

10.2-4.
(a) FALSE. It uses a recursive relationship that enables solving for the optimal policy for
stage 8 given the optimal policy for stage Ð8"Ñ [Feature 7, Section 10.2, p.446].
(b) FALSE. Given the current state, an optimal policy for remaining stages is
independent of the policy decisions adopted in previous stages. Therefore, the optimal
immediate decision depends on only the current state and not on how you got there. This
is the Principle of Optimality for dynamic programming [Feature 5, Section 10.2, p.446].
(c) FALSE. The optimal decision at any stage depends on only the state at that stage and
not on the past. This is again the Principle of Optimality [Feature 5, Section 10.2, p.446].
10.3-1.
The Military Airlift Command (MAC) employed dynamic programming in scheduling its
aircraft, crew and mission support resources during Operation Desert Storm. The primary
goal was to deliver cargo and passengers on time in an environment with time and space
constraints. The missions are scheduled sequentially. The schedule of a mission imposes
resource constraints on the schedules of following missions. A balance among various
objectives is sought. In addition to maximizing timely deliveries, MAC aimed at reducing
late deliveries, total flying time of each mission, ground time and frequency of crew
changes. Maximizing on-time deliveries is included in the model as a lower bound on the
load of the mission. The problem for any given mission is then to determine a feasible
schedule that minimizes a weighted sum of the remaining objectives. The constraints are
the lower bound constraints, crew and ground-support availability constraints. Stages are
the airfields in the network and states are defined as airfield, departure time, and
remaining duty day. The solution of the problem is made more efficient by exploiting the
special structure of the objective function.
The software developed to solve the problems cost around $2 million while the airlift
operation cost over $3 billion. Hence, even a small improvement in efficiency meant a
considerable return on investment. A systematic approach to scheduling yielded better

10-4
coordination, improved efficiency, and error-proof schedules. It enabled MAC not only
to respond quickly to changes in the conditions, but also to be proactive by evaluating
different scenarios in short periods of time.
10.3-2.
Let B8 be the number of crates allocated to store 8, :8 ÐB8 Ñ be the expected profit from
allocating B8 to store 8 and =8 be the number of crates remaining to be allocated to stores
5 8. Then 08‡ Ð=8 Ñ œ max Ò:8 ÐB8 Ñ  08" ‡
Ð=8 B8 ÑÓ. Number of stages: 3
!ŸB8 Ÿ=8

10-5
10.3-3.
Let B8 be the number of study days allocated to course 8, :8 ÐB8 Ñ be the number of grade
points expected when B8 days are allocated to course 8 and =8 be the number of study
days remaining to be allocated to courses 5 8. Then
08‡ Ð=8 Ñ œ max ‡
Ò:8 ÐB8 Ñ  08" Ð=8 B8 ÑÓ.
"ŸB8 ŸminÐ=8 ß%Ñ

Number of stages: 4
=% 0%‡ Ð=% Ñ B‡%
" % "
# % #
$ & $
% ) %
0$ Ð=$ ß B$ Ñ
=$ " # $ % 0$‡ Ð=$ Ñ B‡$
# )    ) 1
$ ) "!   "! #
% * "! ""  "" $
& "# "" "" "$ "$ %
0# Ð=# ß B# Ñ
=# " # $ % 0#‡ Ð=# Ñ B‡#
$ "$    "$ 1
% "& "%   "& "
& "' "' "'  "' "ß #ß $
' ") "( ") "' ") "ß $
0" Ð=" ß B" Ñ
=" " # $ % 0"‡ Ð=" Ñ B‡"
( "* "* #" #" #" $ß %
Optimal Solution B‡" B‡# B‡$ B‡%
" $ " # "
# % " " "

10-6
10.3-4.
Let B8 be the number of commercials run in area 8, :8 ÐB8 Ñ be the number of votes won
when B8 commercials are run in area 8 and =8 be the number of commercials remaining
to be allocated to areas 5 8. Then
08‡ Ð=8 Ñ œ ‡
max Ò:8 ÐB8 Ñ  08" Ð=8 B8 ÑÓ.
!ŸB8 Ÿ=8

Number of stages: 4

10-7
10.3-5.
Let B8 be the number of workers allocated to precinct 8, :8 ÐB8 Ñ be the increase in the
number of votes if B8 workers are assigned to precinct 8 and =8 be the number of
workers remaining at stage 8. Then
08‡ Ð=8 Ñ œ ‡
max Ò:8 ÐB8 Ñ  08" Ð=8 B8 ÑÓ.
!ŸB8 Ÿ=8

Number of stages: 4

10-8
10.3-6.
Let &B8 be the number of jet engines produced in month 8 and =8 be the inventory at the
beginning of month 8. Then 08‡ Ð=8 Ñ is:
‡
min Ò-8 B8  .8 maxÐ=8  B8  <8 ß !Ñ  08" ÐmaxÐ=8  B8  <8 ß !ÑÑÓ
maxÐ<8 =8 ß!ÑŸB8 Ÿ78

and 0%‡ Ð=% Ñ œ -% maxÐ=%  <% ß !Ñ.

Using the following data adjusted to reflect that B8 is one fifth of the actual production,
Month <8 78 -8 .8
" # & &Þ%! !Þ!(&
# $ ( &Þ&& !Þ!(&
$ & ' &Þ&! !Þ!(&
% % # &Þ'& !Þ!(&
the following tables are produced:
=% 0%‡ Ð=% Ñ B‡%
# ""Þ$! #
$ &Þ'& "
% !Þ!! !
0$ Ð=$ ß B$ Ñ
=$ ! " # $ % & ' 0$‡ Ð=$ Ñ B‡$
"       %%Þ%& %%Þ%& '
#      $)Þ*& $)Þ)(& $)Þ)(& '
$     $$Þ%& $$Þ$(& $$Þ$! $$Þ$! '
%    #(Þ*& #(Þ)(& #(Þ)!  #(Þ)! &
&   ##Þ%& ##Þ$(& ##Þ$!   ##Þ$! %
'  "'Þ*& "'Þ)(& "'Þ)!    "'Þ)! $
( ""Þ%& ""Þ$(& ""Þ$!     ""Þ$! #
0# Ð=# ß B# Ñ
=# ! " # $ % & ' ( 0#‡ Ð=# Ñ B#‡
!     ''Þ(#& ''Þ((& ''Þ)#& ''Þ*& ''Þ(#& %
"    '"Þ"(& '"Þ##& '"Þ#(& '"Þ%! '"Þ&#& '"Þ"(& $
#   &&Þ'#& &&Þ'(& &&Þ(#& &&Þ)& &&Þ*(& &'Þ"! &&Þ'#& #
$  &!Þ!(& &!Þ"#& &!Þ"(& &!Þ$! &!Þ%#& &!Þ&& &!Þ'(& &!Þ!(& "
0" Ð=" ß B" Ñ
=" ! " # $ % & 0"‡ Ð=" Ñ B‡"
!   ((Þ&#& ((Þ%& ((Þ$(& ((Þ$! ((Þ$! &
Hence, the optimal production schedule is to produce & † & œ #& units in the first month,
" † & œ & in the second, ' † & œ $! in the third and # † & œ "! in the last month.

10-9
10.3-7.
(a) Let B8 be the amount in million dollars spent in phase 8, =8 be the amount in million
dollars remaining, :" ÐB" Ñ be the initial share of the market attained in phase 1 when B" is
spent in phase 1, and :8 ÐB8 Ñ be the fraction of this market share retained in phase 8 if B8
is spent in phase 8, for 8 œ #ß $. Number of stages: 3
=$ 0$‡ Ð=$ Ñ B‡$
! !Þ$ !
" !Þ& "
# !Þ' #
$ !Þ( $
0# Ð=# ß B# Ñ
=# ! " # $ 0#‡ Ð=# Ñ B‡#
! !Þ!'    !Þ!' !
" !Þ" !Þ"#   !Þ"# "
# !Þ"# !Þ# !Þ"&  !Þ# "
$ !Þ"% !Þ#% !Þ#& !Þ") !Þ#& #
0" Ð=" ß B" Ñ
=" " # $ % 0"‡ Ð=" Ñ B‡"
% & ' %Þ) $ ' #
The optimal solution is B‡" œ #, B‡# œ ", and B‡$ œ ". Hence, it is optimal to spend two
million dollars in phase 1 and one million dollar in each one the phases 2 and 3. This will
result in a final market share of 6%.
(b) Phase 3: = 0$‡ Ð=Ñ B‡$
!Ÿ=Ÿ% !Þ'  !Þ!(= =
Phase 2: 0# Ð=ß B# Ñ œ Ð!Þ%  !Þ"B# ÑÒ!Þ'  !Þ!(Ð=  B# ÑÓ
œ !Þ!(B##  Ð!Þ!(=  !Þ!$#ÑB#  Ð!Þ#%  !Þ!#)=Ñ
`0# Ð=ßB# Ñ = "'
`B# œ !Þ!"%B#  !Þ!!(=  !Þ!$# œ ! Ê B‡# œ #  (

If = Ÿ #=  "'( : B‡# œ = because 0# Ð=ß B# Ñ is strictly increasing on the interval Ò!ß #=  "'
( Ó,
so on Ò!ß =Ó.

10-10
 = "' = "'
If =  #  (: B‡# œ #  ( because then the global maximizer is feasible.
We can summarize this result as:
B‡# Ð=Ñ œ minŠ #=  ( ß = ‹.
"'

$# = "'
Now since ! Ÿ = Ÿ % Ÿ (, =Ÿ #  (, so B‡# Ð=Ñ œ = and 0#‡ Ð=Ñ œ !Þ!'=  !Þ#%.
Phase 1: 0" Ð%ß B" Ñ œ Ð"!B"  B#" ÑÒ!Þ!'Ð%  B" Ñ  !Þ#%Ó
œ !Þ!'B$"  "Þ!)B#"  %Þ)B"
`0" Ð%ßB" Ñ
`B" œ !Þ")B#" #Þ"'B"  %Þ) œ !
#Þ"'„È#Þ"'# %Ð!Þ")ÑÐ%Þ)Ñ
Ê B‡" œ #Ð!Þ")Ñ œ #Þ*%& or *Þ!&&.

The derivative of 0" Ð%ß B" Ñ is nonnegative for B" Ÿ #Þ*%& and B" *Þ!&& and
nonpositive otherwise, so 0" Ð%ß B" Ñ is nonincreasing on the interval Ò#Þ*%&ß *Þ!&&Ó, and
nondecreasing else. Thus, 0" Ð%ß B" Ñ attains its maximum over the interval Ò!ß %Ó at
B‡" œ #Þ*%& with 0"‡ Ð%Ñ œ 'Þ$!#. Accordingly, it is optimal to spend #Þ*%& million dollars
in Phase 1, "Þ!&& in Phase 2 and Phase 3. This returns a market share of 6.302%.
10.3-8.
Let B8 be the number of parallel units of component 8 that are installed, :8 ÐB8 Ñ be the
probability that the component will function if it contains B8 parallel units, -8 ÐB8 Ñ be the
cost of installing B8 units of component 8, =8 be the amount of money remaining in
hundreds of dollars. Then
08‡ Ð=8 Ñ œ max ‡
Ò:8 ÐB8 Ñ08" Ð=8 -8 ÐB8 ÑÑÓ
B8 œ!ßáßminÐ$ß!=8 Ñ

where !=8 ´ maxÖ! À -8 Ð!Ñ Ÿ =8 ß ! integer×.

=% 0%‡ Ð=% Ñ B‡%
!ß " ! !
# !Þ& "
$ !Þ( #
% Ÿ =% Ÿ "! !Þ* $
0$ Ð=$ ß B$ Ñ œ P$ ÐB$ Ñ0%‡ Ð=$ -$ ÐB$ ÑÑ

10-11
 0$ Ð=$ ß B$ Ñ
=$ ! " # $ 0$‡ Ð=$ Ñ B‡$
! !    ! !
"ß # ! !   ! !ß "
$ ! !Þ$& !  !Þ$& "
% ! !Þ%* ! ! !Þ%* "
& ! !Þ'$ !Þ%! ! !Þ'$ "
' ! !Þ'$ !Þ&' !Þ%& !Þ'$ "
( ! !Þ'$ !Þ(# !Þ'$ !Þ(# #
) Ÿ =$ Ÿ "! ! !Þ'$ !Þ(# !Þ)" !Þ)" $
0# Ð=# ß B# Ñ œ P# ÐB# Ñ0$‡ Ð=# -# ÐB# ÑÑ
0# Ð=# ß B# Ñ
=# ! " # $ 0#‡ Ð=# Ñ B‡#
!ß " !    ! !
#ß $ ! !   ! !ß "
% ! ! !  ! !ß "ß #
& ! !Þ#"! ! ! !Þ#"! "
' ! !Þ#*% ! ! !Þ#*% "
( ! !Þ$() !Þ#%& ! !Þ$() "
) ! !Þ$() !Þ$%$ !Þ#)! !Þ$() "
* ! !Þ%$# !Þ%%" !Þ$*# !Þ%%" #
"! ! !Þ%)' !Þ%%" !Þ&!% !Þ&!% $
0" Ð=" ß B" Ñ œ P" ÐB" Ñ0#‡ Ð=" -" ÐB" ÑÑ
0" Ð=" ß B" Ñ
=" ! " # $ 0"‡ Ð=" Ñ B‡"
"! ! !Þ## !Þ##( !Þ$!# !Þ$!# $
The optimal solution is B‡" œ $, B‡# œ ", B‡$ œ " and B‡% œ $, yielding a system reliability
of 0.3024.
10.3-9.
The stages are 8 œ "ß # and the state is the amount of slack remaining in the constraint,
the goal is to find 0"‡ Ð%Ñ.
=# 0#‡ Ð=# Ñ B‡# 0" Ð=" ß B" Ñ
! ! ! =" ! " # $ % 0"‡ Ð=" Ñ B‡"
" ! ! % "# ' ) ! "' "# !
# % "
$ % "
% "# #
The optimal solution is B‡" œ ! and B‡# œ #.

10-12
10.3-10.
The stages are 8 œ "ß #ß $ and the state is the slack remaining in the constraint, the goal is
to find 0"‡ Ð#!Ñ.
=$ 0$‡ Ð=$ Ñ B‡$ 0# Ð=# ß B# Ñ
!% ! ! =# ! " # 0#‡ Ð=# Ñ B‡#
&* #! " !% !   ! !
"!  "% %! # &' #!   #! !
"&  "* '! $ (* #! $!  $! "
#! )! % "!  "" %! $!  %! !
"#  "$ %! &!  &! "
"% %! &! '! '! #
"&  "' '! &! '! '! !ß #
"(  ") '! (! '! (! "
"* '! (! )! )! #
#! )! (! )! )! !ß #
0" Ð=" ß B" Ñ
=" ! " # $ % & ' 0"‡ Ð=" Ñ B‡"
#! )! "!! ""' "") "#' "$! "#! "$! &
The optimal solution is B‡" œ &, B‡# œ !, B‡$ œ " with an objective value D ‡ œ "$!.
10.3-11.
Let =8 denote the slack remaining in the constraint.
0#‡ Ð=# Ñ œ max ˆ$'B#  $B#$ ‰
Ú  ! for ! Ÿ B  #
!ŸB# Ÿ=#

*B## Û Ê B‡# œ œ
#
`0# Ð=ßB# Ñ =# for ! Ÿ =#  #
Ü  ! for B#  #
`B# œ $'  œ ! for B# œ #
# for # Ÿ =# Ÿ $

0"‡ Ð$Ñ œ max Ò$'B"  *B#"  'B$"  0#‡ Ð$B" ÑÓ

Ú max Ò$'B  *B#  'B$  %)Ó
!ŸB" Ÿ$

œ max Û
" " "
!ŸB" Ÿ"

Ü "ŸB
max Ò$'B"  *B#"  'B"$  $'Ð$B" Ñ  $Ð$B" Ñ$ Ó
Ú
" Ÿ$

Ý
Ý
Ý Ú
")ÐB#"  B"  #Ñ  ! for ! Ÿ B" Ÿ " Ê B"max œ "
Ý  ! for " Ÿ B"  #  È"$
Û È
Ý
Ý *ÐB#"  %B"  *ÑÛ œ ! for B" œ #  È"$ Ÿ Ê B" œ #  "$
`0" Ð$ßB" Ñ
œ
Ý Ý
`B" max

Ü Ü  ! for B"  #  È"$

Since 0" Ð$ß "Ñ  0" Ð$ß #  È"$Ñ, B‡" œ #  È"$ ¶ "Þ'" and B‡# œ &  È"$ ¶ "Þ$*
with the optimal objective value being 0"‡ Ð$Ñ ¶ *)Þ#$.

10-13
10.3-12.
08‡ Ð=8 Ñ œ min Ò"!!ÐB8  =8 Ñ#  #!!!ÐB8  <8 Ñ  08"
‡
ÐB8 ÑÓ
<8 ŸB8 Ÿ#&&

8 œ %:
=% 0%‡ Ð=% Ñ B‡%
#!! Ÿ =% Ÿ #&& "!!Ð#&&  =% Ñ# #&&
8 œ $: 0$ Ð=$ ß B$ Ñ œ "!!ÐB$  =$ Ñ#  #!!!ÐB$  #!!Ñ  "!!Ð#&&  B$ Ñ#
` 0$ Ð=$ ßB$ Ñ
`B$ œ #!!ÐB$  =$ Ñ  #!!!  #!!Ð#&&  B$ Ñ
=$ #%&
œ #!!Ò#B$  Ð=$  #%&ÑÓ œ ! Ê B$ œ #

If "&& Ÿ =$ Ÿ #'&, #!! Ÿ =$ #%&

# Ÿ #&&, so B$ œ =$ #%&
# is feasible for #%! Ÿ =$ Ÿ #&&
‡ # #
and 0$ Ð=$ Ñ œ #&Ð#%&  =$ Ñ  #&Ð#'&  =$ Ñ  "!!!Ð=$  "&&Ñ.
=$ 0$‡ Ð=$ Ñ B‡$
=$ #%&
#%! Ÿ =$ Ÿ #&& #&Ð#%&  =$ Ñ#  #&Ð#'&  =$ Ñ#  "!!!Ð=$  "&&Ñ #

8 œ #: 0# Ð=# ß B# Ñ œ "!!ÐB#  =# Ñ#  #!!!ÐB#  #%!Ñ  0$‡ ÐB# Ñ

` 0# Ð=# ßB# Ñ
`B# œ #!!ÐB#  =# Ñ  #!!!  &!Ð#%&  B# Ñ  &!Ð#'&  B# Ñ  "!!!
#=# ##&
œ "!!Ò$B#  Ð#=#  ##&ÑÓ œ ! Ê B# œ $
#=# ##& #=# ##&
If #%(Þ& Ÿ =# Ÿ #&&, #%! Ÿ $ Ÿ #&&, so B‡# œ $ and

0#‡ Ð=# Ñ œ "!!Š #=# ##&  =# ‹  #!!!Š #=# ##&  #%!‹  0$‡ Š #=# ##& ‹
#
$ $ $
"!!
œ * ÒÐ##&  =# Ñ#  Ð#&&  =# Ñ#  Ð#)&  =# Ñ#  '!Ð$=#  '"&ÑÓ.
#=# ##& ` 0# Ð=# ßB# Ñ
If ##! Ÿ =# Ÿ #%(Þ&, $ Ÿ #%! Ÿ B# , so `B# ! and hence B‡# œ #%! and
0#‡ Ð=# Ñ œ "!!Ð#%!  =# Ñ#  #!!!Ð#%!  #%!Ñ  0$‡ Ð#%!Ñ œ "!!Ð#%!  =# Ñ#  "!"ß #&!.
=# 0#‡ Ð=# Ñ B‡#
##! Ÿ =# Ÿ #%(Þ& "!!Ð#%!  =# Ñ#  "!"ß #&! #%!
"!! # # # #=# ##&
#%(Þ& Ÿ =# Ÿ #&& * ÒÐ##&  =# Ñ  Ð#&&  =# Ñ  Ð#)&  =# Ñ  '!Ð$=#  '"&ÑÓ $

8 œ ": 0" Ð#&&ß B" Ñ œ "!!ÐB"  #&&Ñ#  #!!!ÐB"  ##!Ñ  0#‡ ÐB" Ñ
If ##! Ÿ B" Ÿ #%(Þ&:
` 0# Ð#&&ßB" Ñ
`B" œ #!!ÐB"  %)&Ñ œ ! Ê B‡" œ #%#Þ&.
If #%(Þ& Ÿ B" Ÿ #&&:
` 0# Ð#&&ßB" Ñ )!!
`B" œ $ ÐB"  #%!Ñ  ! Ê B‡" œ #%(Þ&.
The optimal solution is B‡" œ #%#Þ& and
0"‡ Ð#&&Ñ œ "!!Ð#%#Þ&  #&&Ñ#  #!!!Ð#%#Þ&  ##!Ñ  0#‡ Ð#%#Þ&Ñ œ "'#ß &!!.

10-14
 =" 0"‡ Ð=" Ñ B‡"
#&& "'#ß &!! #%#Þ&
Summer Autumn Winter Spring
#%#Þ& #%! #%#Þ& #&&
10.3-13.
Let =8 be the amount of the resource remaining at beginning of stage 8.
8 œ $: max Ð%B$  B#$ Ñ
!ŸB$ Ÿ=$
`
`B$ Ð%B$  B#$ Ñ œ %  #B$ œ ! Ê B‡$ œ #
`#
`B#$
Ð%B$  B#$ Ñ œ #  ! Ê B‡$ œ # is a maximum.

=$ 0$* Ð=$ Ñ B*$

! Ÿ =$ Ÿ # %=$  =#$ =$
# Ÿ =$ Ÿ % % #
8 œ #: max Ò#B#  0$* Ð=#  B# ÑÓ
!ŸB# Ÿ=#

If ! Ÿ =#  B# Ÿ #: max Ò#B#  %Ð=#  B# Ñ  Ð=#  B# Ñ# Ó

!ŸB# Ÿ=#
`
`B# Ò#B#  %Ð=#  B# Ñ  Ð=#  B# Ñ# Ó œ #  #=#  #B# œ ! Ê B#‡ œ =#  "
`#
`B##
Ò#B#  %Ð=#  B# Ñ  Ð=#  B# Ñ# Ó œ #  ! Ê B#‡ œ =#  " is a maximum.

0#* Ð=# Ñ œ #=#  ".

If # Ÿ =#  B# Ÿ %: max Ð#B#  %Ñ, B‡# œ =#  # and 0#* Ð=# Ñ œ #=#  #=#  ".
!ŸB# Ÿ=#

=# 0#* Ð=# Ñ B*#

! Ÿ =# Ÿ " %=#  =## !
" Ÿ =# Ÿ % #=#  " =#  "
8 œ ": max Ò#B#"  0#* Ð%  #B" ÑÓ
!ŸB" Ÿ="

If ! Ÿ %  #B" Ÿ ": max Ò#B#"  %Ð%  #B" Ñ  Ð%  #B" Ñ# Ó œ Ð#B#"  )B" Ñ

!ŸB" Ÿ="
` #
`B" Ð#B"  )B" Ñ œ %B"  ) œ ! Ê B‡" œ #
`#
`B#"
Ð#B#"  )B" Ñ œ %  ! Ê B‡" œ # is a maximum.

0" Ð%ß #Ñ œ ).
If " Ÿ %  #B" Ÿ %: max Ò#B#"  #Ð%  #B" Ñ  "Ó œ Ð#B#"  %B"  *Ñ
!ŸB" Ÿ="
` #
`B" Ð#B"  %B"  *Ñ œ %B"  % œ ! Ê B" œ "

10-15
 `#
`B#"
Ð#B#"  %B"  *Ñ œ %  ! Ê B" œ " is a minimum.

Corner points: " œ %  #B" Ê B" œ $Î#, 0" Ð%ß $Î#Ñ œ (Þ&
% œ %  #B" Ê B" œ !, 0" Ð%ß !Ñ œ * is maximum.
Hence, B‡" œ !, B‡# œ $, B‡$ œ " and 0"‡ Ð%Ñ œ *.
10.3-14.
8 œ #: min #B## Ê B‡# œ È=# and 0#‡ Ð=# Ñ œ #=# ,
B## =#

where =# represents the amount of # used by B## .

8 œ ": minÒB%"  0#‡ ÐÐ#  B#" Ñ ÑÓ œ ÒB"%  #Ð#  B"# Ñ Ó,
B"

where Ð#  B#" Ñ œ maxÖ!ß #  B"# ×.

`
If B#" Ÿ #: %
`B" ÐB"  %  #B#" Ñ œ %B$"  %B" œ ! Ê B" œ !ß "ß ".
`#
`B#"
ÐB%"  %  #B#" Ñ œ "#B#"  %
`#
B" œ !, `B#"
ÐB%"  %  #B#" Ñ œ %  !, so B" œ ! is a local maximum.
`#
B" œ "ß ", `B#"
ÐB%"  %  #B#" Ñ œ )  !, so B" œ "ß " are local minima
with D œ $.
If B#" #: B" œ ! and D œ %  $.
Hence, ÐB‡" ß B‡# Ñ − ÖÐ"ß "Ñß Ð"ß "Ñß Ð"ß "Ñß Ð"ß "Ñ×, all with D ‡ œ $.
10.3-15.
(a) Let =8 − Ö"ß #ß %× be the remaining factor % entering stage 8.
8 œ $: 8 œ #:
=$ 0$* Ð=$ Ñ B‡$ 0# Ð=# ß B# Ñ
" "' " =# " # % 0#‡ Ð=# Ñ B‡#
# $# # " #!   #! "
% '% $ # $' $#  $' "
% ') %) )! )! %
8 œ ":
0" Ð=" ß B" Ñ
=" " # % 0"‡ Ð=" Ñ B‡"
% )" %% )% )% %
The optimal solution is ÐB‡" ß B‡# ß B‡$ Ñ œ Ð%ß "ß "Ñ with D ‡ œ )%.
(b) As in part (a), let =8 be the remaining factor (not necessarily integer) at stage 8.
0$‡ Ð=$ Ñ œ "'=$ and B‡$ œ =$

10-16
0#‡ Ð=# Ñ œ max Ö%B##  0$‡ Ð=# ÎB# Ñ× œ max Ö%B##  "'=# ÎB# ×
"ŸB# Ÿ=# "ŸB# Ÿ=#
` 0# Ð=# ßB# Ñ ` # 0# Ð=# ßB# Ñ
`B# œ %B#  "'=# ÎB## and `B##
œ %  $#=# ÎB#$  !

when =# , B# !. Thus 0# Ð=# ß B# Ñ is convex in B# when =# , B# !. The maximum should

occur at one of the endpoints.
B# œ ", 0# Ð=# ß "Ñ œ %  "'=#
B# œ =# , 0# Ð=# ß =# Ñ œ %=##  "'
%  "'=# %=##  "' Í Ð=#  $ÑÐ=#  "Ñ Ÿ ! Í " Ÿ =# Ÿ $

B‡# œ œ and 0#‡ Ð=# Ñ œ œ #

" if " Ÿ =# Ÿ $ %  "'=# if " Ÿ =# Ÿ $
=# if $ Ÿ =# Ÿ % %=#  "' if $ Ÿ =# Ÿ %
0"‡ Ð=" Ñ œ max ÖB"$  0#‡ Ð%ÎB" Ñ×
"ŸB" Ÿ%

œ maxœ max šB$"  %Š "'

B#"
‹  "'›ß max šB$"  %  "'Š B%" ‹›
"ŸB" Ÿ%Î$ %Î$ŸB" Ÿ%

`#
`B#"
šB$"  %Š "'
B#
‹  "'› œ 'B"  #!%ÎB"%  ! when B" !
"

`#
`B#"
šB$"  %  "'Š B%" ‹› œ 'B"  "#)ÎB"#  ! when B" !

Hence, the maximum occurs at an endpoint.

B" œ ", 0" Ð=" ß "Ñ œ )"
B" œ %Î$, 0" Ð=" ß %Î$Ñ ¸ &%Þ$(
B" œ %, 0" Ð=" ß %Ñ œ )%
0"‡ Ð=" Ñ œ maxÖ)"ß &%Þ$(ß )%× œ )% and ÐB‡" ß B‡# ß B‡$ Ñ œ Ð%ß "ß "Ñ, just as when the
variables are restricted to be integers.
10.3-16.
Let =8 be the slack remaining in the constraint B"  B#  B$ Ÿ ", entering the 8th stage.
0$‡ Ð=$ Ñ œ max B$ œ =$ and B‡$ œ =$
!ŸB$ Ÿ=$

0#‡ Ð=# Ñ œ max ÖÐ"  B# Ñ0$‡ Ð=#  B# Ñ× œ max ÖÐ"  B# ÑÐ=#  B# Ñ×

=
# ŸB# =# ŸB#

where =
# œ maxÖ=# ß !×.
` 0# Ð=# ßB# Ñ
`B# œ #B#  Ð=#  "Ñ œ ! Ê B# œ Ð"  =# ÑÎ#
` # 0# Ð=# ßB# Ñ
`B##
œ #  !, so 0# Ð=# ß B# Ñ is concave in B# .

B# œ Ð=#  "ÑÎ#, 0# Ð=# ß Ð"  =# ÑÎ#Ñ œ Ð"  =# Ñ# Î%

# , 0# Ð=# ß =# Ñ œ œ
! if =# Ÿ !
B# œ = 
=# if =# !

10-17
Ð"  =# Ñ# Î# maxÖ!ß =# ×
B# œ Ð"  =# ÑÎ# is feasible if and only if =
# Ÿ Ð"  =# ÑÎ#, equivalently when =# ".

0#‡ Ð=# Ñ œ œ œ
! if =# Ÿ " ‡ =
# œ =# if =# Ÿ "
# and B# œ
Ð"  =# Ñ Î% if =# " Ð"  =# ÑÎ# if =# "

maxœ max šB" Š %"  Ð"  B" Ñ‹›ß !

B#
0"‡ Ð=" Ñ œ maxÖB" 0#‡ Ð"  B" Ñ× œ
B" ! !ŸB Ÿ# "

max š %  B#"  B" ›

B$"
œ
!ŸB" Ÿ#

#„È%$
`B" š %  B#"  B" › œ
$
` B" $B#" % #
%  #B"  " œ ! Ê B" œ $Î# œ $ „ $

Hence, ÐB‡" ß B‡# ß B‡$ Ñ œ Ð#Î$ß "Î$ß #Î$Ñ and D ‡ œ )Î#(.

10.3-17.
Let = œ ÐV" ß V# Ñ, where V3 is the slack in the 3th constraint.
8 œ #: 0# ÐV" ß V# ß B# Ñ œ #B# , ! Ÿ B# Ÿ minÖV" Î#ß V# ×
= 0#‡ Ð=Ñ B‡#

ŒV 
V"
"!minÖV" Î#ß V# × minÖV" Î#ß V# ×
#

8 œ ": 0" Ð'ß )ß B" Ñ œ "&B"  0#‡ Ð'  B" ß )  $B" Ñ

œ "&B"  "!minÖÐ'  B" ÑÎ#ß )  $B" ×, for ! Ÿ B" Ÿ )Î$

œœ
"!B"  $! if ! Ÿ B" Ÿ #
)!  "&B" if # Ÿ B" Ÿ )Î$

max 0" Ð'ß )ß B" Ñ œ maxœ max 0" Ð'ß )ß B" Ñß max 0" Ð'ß )ß B" Ñ œ &!
!ŸB" Ÿ)Î$ !ŸB" Ÿ# #ŸB" Ÿ)Î$

and B‡" œ #.
The optimal solution is ÐB‡" ß B‡# Ñ œ Ð#ß #Ñ and D ‡ œ &!.

10-18
10.3-18.
Let = œ ÐV" ß V# Ñ, where V3 is the slack in the 3th constraint.

0$ ÐV" ß V# ß B$ Ñ œ œ
! if B$ œ !
"  B$ if B$  !

0$‡ ÐV" ß V# Ñ œ maxœ!ß max Ð"  B$ Ñ œ maxÖ!ß "  ÐV" Î#Ñ×

!ŸB$ ŸV" Î#

œœ
"  ÐV" Î#Ñ if ! Ÿ V" Ÿ #
! if V" #

B‡$ œ œ "
V Î# if ! Ÿ V" Ÿ #
! if V" #
0# ÐV" ß V# ß B# Ñ œ (B#  0$‡ ÐV"  $B# ß V# Ñ

œœ
(B#  "  ÐV"  $B# ÑÎ# if ! Ÿ V"  $B# Ÿ #
(B# if V"  $B# #

0#‡ ÐV" ß V# Ñ œ max œ!ß max Ð"  B$ Ñ œ maxÖ!ß "  ÐV" Î#Ñ×
Ú
!ŸB# ŸminÖV" Î$ßV# × !ŸB$ ŸV" Î#

Ý (V$ " if V$" Ÿ V#

œ Û (V#
Ý "(V#
if V"$# Ÿ V# Ÿ V"

Ü # "
$
V"
if V# Ÿ V"$#
Ú
#

Ý V$" if V$" Ÿ V#
B‡# œ Û V#
Ý
if V"$# Ÿ V# Ÿ V"

Ü V#
$
if V# Ÿ V"$#
0"‡ Ð'ß &Ñ œ max Ò$B"  0#‡ Ð'  B" ß &  B" ÑÓ
!ŸB" Ÿ&

œ maxœ max ’$B"  (Ð'B" Ñ

$ “ß max Ò$B"  (Ð&  B" ÑÓ
!ŸB" Ÿ*Î# *Î#ŸB" Ÿ&

œ maxœ max ’ #B$"  "%“ß max Ò$&  #B" Ó œ "(

!ŸB" Ÿ*Î# *Î#ŸB" Ÿ&

The optimal solution is ÐB‡" ß B‡# ß B‡$ Ñ œ Š *# ß "# ß !‹ and D ‡ œ "(.

10.4-1.
Let =8 be the current fortune of the player, E be the event to have $100 at the end and \8
be the amount bet at the 8th match.
0$‡ Ð=$ Ñ œ max ÖPÖEl=$ ××
!ŸB$ Ÿ=$

! Ÿ =$  &!, 0$‡ Ð=$ Ñ œ !.

10-19
&! Ÿ =$  "!!, 0$‡ Ð=$ Ñ œ œ
! if B‡$ Á "!!  =$
"Î# if B‡$ œ "!!  =$

=$ œ "!!, 0$‡ Ð=$ Ñ œ œ

! if B‡$  !
" if B‡$ œ !

=$  "!!, 0$‡ Ð=$ Ñ œ œ

! if B‡$ Á =$  "!!
"Î# if B‡$ œ =$  "!!
=$ 0$‡ Ð=$ Ñ B‡$
! Ÿ =$  &! ! ! Ÿ B‡$ Ÿ &!
&! Ÿ =$  "!! "Î# "!!  =$
=$ œ "!! " !
"!!  =$ "Î# =$  "!!

0#‡ Ð=# Ñ œ max ’ " 0$‡ Ð=#  B# Ñ  "# 0$‡ Ð=#  B# Ñ“

!ŸB# Ÿ=# #

=# 0#‡ Ð=# Ñ B‡#

! Ÿ =#  #& ! ! Ÿ B# Ÿ =#
#& Ÿ =#  &! ! ! Ÿ B# Ÿ &!  =#
"Î% &!  =# Ÿ B# Ÿ =#
=# œ &! "Î% ! Ÿ B#  &!
"Î# B# œ &!
&!  =#  (& "Î# ! Ÿ B#  =#  &!
"Î% =#  &!  B#  "!!  =#
"Î# B# œ "!!  =#
"Î% "!!  =#  B# Ÿ =#
=# œ (& "Î# ! Ÿ B#  #&
$Î% B# œ #&
"Î% #& Ÿ B# Ÿ (&
(&  =#  "!! "Î# ! Ÿ B#  "!!  =#
$Î% B# œ "!!  =#
"Î# "!!  =#  B# Ÿ =#  &!
"Î% =#  &!  B# Ÿ =#
=# œ "!! " B# œ !
"Î# !  B# Ÿ &!
"Î% &! Ÿ B# Ÿ "!!
"!!  =# "Î# ! Ÿ B# Ÿ =#  "!!
$Î% B# œ =#  "!!
"Î# =#  "!!  B# Ÿ =#  &!
"Î% =#  &!  B# Ÿ =#
The entries in bold represent the maximum value in each case.
0"‡ Ð(&Ñ œ max ’ " 0#‡ Ð(&  B" Ñ  "# 0#‡ Ð(&  B" Ñ“
!ŸB" Ÿ(& #

10-20
 Ú
Ý
Ý
Ý
Ý &Î)
$Î% if B" œ !

0" Ð(&ß B" Ñ œ Û $Î%

if !  B"  #&
Ý
Ý
if B" œ #&
Ý
Ý "Î#
Ü $Î)
if #&  B" Ÿ &!
if &!  B" Ÿ (&
=" 0"‡ Ð=" Ñ B‡"
(& $Î% ! or #&
Policy B" won "st bet lost "st bet won #nd bet lost #nd bet
" ! #& #& ! &!
# #& ! &! ! !
10.4-2.
(a) Let B8 − Ö!ß Eß F× be the investment made in year 8, =8 be the amount of money on
hand at the beginning of year 8 and 08 Ð=8 ß B8 Ñ be the maximum expected amount of
money by the end of the third year given =8 and B8 .
‡
For ! Ÿ =8  "!ß !!!, since one cannot invest less than $"!ß !!!, 08 Ð=8 ß B8 Ñ œ 08" Ð=8 Ñ
‡
and B8 œ !.
For =8 "!ß !!!,
Ú 0 ‡ Ð=8 Ñ
08 Ð=8 ß B8 Ñ œ Û !Þ#&08" Ð=8  "!ß !!!Ñ  !Þ(&08" Ð=8  "!ß !!!Ñ if B8 œ E
8" if B8 œ !

Ü !Þ*08"
‡ ‡
‡ ‡
Ð=8 Ñ  !Þ"08" Ð=8  "!ß !!!Ñ if B8 œ F
=$ 0$‡ Ð=$ Ñ B‡$
! Ÿ =$  "!ß !!! =$ !
=$ "!ß !!! =$  &ß !!! E
0# Ð=# ß B# Ñ
=# ! E F 0#‡ Ð=# Ñ B‡#
! Ÿ =#  "!ß !!! =#   =# !
"!ß !!! Ÿ =#  #!ß !!! =#  &ß !!! =#  )ß (&! =#  'ß !!! =#  )ß (&! E
=# #!ß !!! =#  &ß !!! =#  "!ß !!! =#  'ß !!! =#  "!ß !!! E
0" Ð=" ß B" Ñ
=" ! E F 0"‡ Ð=" Ñ B‡"
"!ß !!! ")ß (&! ##ß &!! "*ß )(& ##ß &!! F
The optimal policy is to invest in E as long as there is enough money. The expected
fortune after three years using this strategy is $##,&!!.
(b) Let B8 and =8 be defined as in (a). Let 08 Ð=8 ß B8 Ñ be the maximum probability of
having at least $#0,000 after 3 years given =8 and B8 .

10-21
 0$ Ð=$ ß B$ Ñ
=$ ! E F 0$‡ Ð=$ Ñ B‡$
! Ÿ =$  "!ß !!! !   ! !
"!ß !!! Ÿ =$  #!ß !!! ! !Þ(& !Þ" !Þ(& E
#!ß !!! Ÿ =$  $!ß !!! " !Þ(& " " !ß F
=$ $!ß !!! " " " " !ß Eß F
0# Ð=# ß B# Ñ
=# ! E F 0#‡ Ð=# Ñ B‡#
! Ÿ =#  "!ß !!! !   ! !
"!ß !!! Ÿ =#  #!ß !!! !Þ(& !Þ(& !Þ((& !Þ((& F
=# #!ß !!! " !Þ(& " " !ß F
0" Ð=" ß B" Ñ
=" ! E F 0"‡ Ð=" Ñ B‡"
"!ß !!! !Þ((& !Þ(& !Þ(*(& !Þ(*(& F
With this objective, there is a number of optimal policies. The optimal action in the first
period is to invest in F . If the return from it is only $"!ß !!!, one is indifferent between
investing in F or not investing at all in the second year. Depending on the second year's
investment choice and its return, third year's starting budget can be either $"!ß !!!,
$#!ß !!! or $$!ß !!!. If it is $"!ß !!!, then it is best to invest it in E. If it is $#!ß !!!,
investing in F or not investing are best. Finally if it is $$!ß !!!, anything is optimal,
since $#!ß !!! is guaranteed. Using this policy, the probability of having at least $#!ß !!!
by the end of the third year is !Þ(*(&.
10.4-3.
08 Ð"ß B8 Ñ œ OÐB8 Ñ  B8  Š "$ ‹ 08" Ð"Ñ  ’"  Š "$ ‹ “08"
B8 B8
‡ ‡
Ð!Ñ

œ OÐB8 Ñ  B8  Š "$ ‹ 08"

B8
‡
Ð"Ñ

since 08‡ Ð!Ñ œ ! for every 8. 0$‡ Ð"Ñ œ "', 0$‡ Ð!Ñ œ ! and OÐB8 Ñ œ ! if B8 œ !,
OÐB8 Ñ œ $ if B8  !.
0# Ð=# ß B# Ñ
=# ! " # $ % 0#‡ Ð=# Ñ B‡#
! !     ! !
" "' *Þ$$ 'Þ() 'Þ&* (Þ#! 'Þ&* $
0" Ð=" ß B" Ñ
=" ! " # $ % 0"‡ Ð=" Ñ B"‡
" 'Þ&* 'Þ#! &Þ($ 'Þ#% (Þ!) &Þ($ #
The optimal policy is to produce two in the first run and to produce three in the second
run if none of the items produced in the first run is acceptable. The minimum expected
cost is $&(3.

10-22
10.4-4.
08‡ Ð=8 Ñ œ maxš "$ 08"
‡
Ð=8  B8 Ñ  #$ 08"
‡
Ð=8  B8 Ñ›,
B8 !

with 0'‡ Ð=' Ñ œ ! for ='  & and 0'‡ Ð=' Ñ œ " for =' &.
=& 0&‡ Ð=& Ñ B&‡
! ! !
" ! !
# ! !
$ #Î$ B‡& #
% #Î$ B‡& "
=& & " B‡& Ÿ =&  &
0% Ð=% ß B% Ñ
=% ! " # $ % 0%‡ Ð=% Ñ B‡%
! !     ! !
" ! !    ! !
# ! %Î* %Î*   %Î* "ß #
$ #Î$ %Î* #Î$ #Î$  #Î$ !Þ#ß $
% #Î$ )Î* #Î$ #Î$ #Î$ )Î* "
=% & "     " B‡% Ÿ =%  &
0$ Ð=$ ß B$ Ñ
=$ ! " # $ % 0$‡ Ð=$ Ñ B‡$
! !     ! !
" ! )Î#(    )Î#( "
# %Î* %Î* "'Î#(   "'Î#( #
$ #Î$ #!Î#( #Î$ #Î$  #!Î#( "
% )Î* )Î* ##Î#( #Î$ #Î$ ##Î#( !ß "
=$ & "     " B‡$ Ÿ =$  &
0# Ð=# ß B# Ñ
=# ! " # $ % 0#‡ Ð=# Ñ B#‡
! !     ! !
" )Î#( $#Î)"    $#Î)" "
# "'Î#( %)Î)" %)Î)"   %)Î)" !ß "ß #
$ #!Î#( '%Î)" '#Î)" #Î$  '%Î)" "
% #%Î#( (%Î)" (!Î)" '#Î)" #Î$ (%Î)" "
=# & "     " B‡# Ÿ =#  &
0" Ð=" ß B" Ñ
=" ! " # 0"‡ Ð=" Ñ B‡"
# %)Î)" "'!Î#%$ "#%Î#%$ "'!Î#%$ "
The probability of winning the bet using the policy given above is "'!Î#%$ œ !Þ'&).

10-23
10.4-5.
Let B8 − ÖEß H× denote the decision variable of quarter 8 œ "ß #ß $, where E and H
represent advertising or discontinuing the product respectively. Let =8 be the level of
sales (in millions) above (=8 !) or below (=8 Ÿ !) the break-even point for quarter
Ð8  "Ñ. Let 08 Ð=8 ß B8 Ñ represent the maximum expected discounted profit (in millions)
from the beginning of quarter 8 onwards given the state =8 and decision B8 .
" ' ,8 " ' ,8 ‡
08 Ð=8 ß B8 Ñ œ $!  &’=8  ,8 +8 +8 >.>“  ,8 +8 +8 08" Ð=8  >Ñ>.>,

where +8 and ,8 are given in the table that follows.

8 +8 ,8
" " &
# ! %
$ " $
For " Ÿ 8 Ÿ $,
" ' ,8 ‡
08 Ð=8 ß EÑ œ $!  &’=8  # “
+8 ,8
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10-25
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of lesion causing this symptom, some of which have been
reproduced in our table. We will not go into any details as to the
character of this symptom, referring the reader to the sources
indicated. In the first case given in our table (Case 10) the
hemianopsia was produced by a tumor in front of, and impinging
upon, the optic chiasm; in the other four cases (Cases 40, 41, 42,
and 43) the tumor was situated in the occipital lobe, and was
surrounded by an area of destroyed tissue. Hemianopsia is not,
strictly speaking, a symptom of brain tumor, but is likely to be present
in cases occurring in certain regions of the brain. Starr's conclusions
with reference to lateral homonymous hemianopsia when it is not
produced by a lesion of one optic tract are that it may result from a
lesion situated either (1) in the pulvinar of one optic thalamus; (2) in
the posterior part of one interior capsule or its radiation backward
toward the occipital lobe; (3) in the medullary portion of the occipital
lobe; or (4) in the cortex of one occipital lobe. The conclusions of
Seguin are only different in so far as they more closely limit the
position of the lesion.
25 Vol. IV.

26 Pp. 84, 85 of present Volume.

27 Amer. Journ. Med. Sci., N. S., vol. lxxxvii., January, 1884, p. 65.

Phosphenes, or subjective sensations of light, occur in various forms

—simply flashes or sheets of light, scintillations, balls of fire, etc.
They are not very common as isolated phenomena, and probably
are dependent in most cases upon irritation of the nerve and retina in
some of the stages of neuro-retinitis. Even visual hallucinations are
occasionally present, as in one of Bennett's cases of tumor of the
Rolandic region.

Conjugate deviation of the eyes, with rotation of the head, a

symptom of the early stages of apoplectic attacks, is also sometimes
observed in brain tumor. The patient is found with both eyes turned
to one side and slightly upward, as if looking over one or the other
shoulder, the head and neck being usually rotated in the same
direction. Sometimes the deviation is slight, sometimes it is marked.
Frequently the muscles of the neck on one side are rigid. The eyes
are commonly motionless, but occasionally exhibit oscillations. This
sign, well known to neurologists, usually disappears in cases of
apoplexy in a few hours or days, although it occasionally persists for
a long time. It will be more fully considered under Local Diagnosis.

Diminution or loss of hearing, tinnitus, and hyperæsthesia of hearing

are all occasionally observed. The most decided disturbances of this
sense are those which are found in connection with tumors of the
base or of the cerebellum in such a position as to involve the
auditory nerve or auditory tracts. Tinnitus, acoustic hyperæsthesia,
with complete or partial deafness, accompanying facial paralysis,
with or without paresis of the limbs of the opposite side, indicate
clearly a tumor of the base so situated as to involve the superficial
origin or intracranial course of the auditory and facial nerves.

The sense of smell is affected, of course, when the olfactory bulbs

are involved in the growth, either directly or by pressure, as in certain
tumors of the antero-frontal region (Cases 4 and 8). Disturbances in
the power of consciously perceiving odors, or abnormal perceptions
of odors or hallucinations of smell, are sometimes present in cerebral
tumors involving certain convolutions. The lower postero-parietal
region or the temporo-sphenoidal region of the base would seem,
from the few reported cases, to be implicated when this sense is
centrally affected. Smell was lost or impaired in two cases of tumors
of the postero-parietal region, in one limited to the supramarginal
convolutions. In a case reported by Allan McLane Hamilton (Case
47), an induration of the lower part of the right temporo-sphenoidal
lobe involving the uncinate gyrus, the patient, preceding light
epileptic attacks, always had an olfactory aura of a peculiar
character—a disagreeable odor, sometimes of smoke and
sometimes of a fetid character. In this case the olfactory nerves were
examined and found to be healthy.

Taste may be involved in several ways. In the first place, subjective

sensations of taste, particularly the so-called metallic taste, may be
present when the growths involve the cranial nerves in such a way
as to cause irritation to be conveyed to the nucleus of the
hypoglossal. When it is remembered that a mild galvanic current
applied to the nape of the neck or face will often cause this metallic
taste, it can be seen that the irritation of a tumor situated at almost
any point of the base might lead to abnormal taste-phenomena.
Neoplasms involving the trunk of the portio dura may of course
cause diminution or loss of taste on the anterior extremity of the
tongue by the involvement of the chorda tympani nerve. In the very
few cases in which the hypoglossal trunk may be involved
disturbances of taste posteriorly may occur. In two cases (Cases 33
and 36) some possible indications as to the cortical areas of taste
are given. One was a tumor so situated as to cause pressure on the
orbital, and possibly anterior, portion of the temporo-sphenoidal lobe;
the other was a lesion closely localized to the supramarginal lobule.

Trophic disturbances of decided character are sometimes present in

cases of brain tumor. Their presence, character, and extent depend
upon the position of the tumor and the cranial nerves involved.
Trophic disorders of the eye have been noted in cases of tumor of
the antero-frontal region, and also of various positions at the base,
especially those so situated as to involve the trigeminal nerve. In a
fibroma of the superior antero-frontal region (Case 1) conjunctivitis
and corneitis of the left eye, with anæsthesia of the conjunctiva, were
present, and were very marked symptoms. This patient, who was
under the care of one of us at the Philadelphia Hospital, was
examined by O. E. Shakespeare, ophthalmologist to the hospital. At
his first examination the bulbar conjunctivæ were slightly injected
and the cornea clear. The sensibility of the cornea was possibly a
little lowered. Ten days later, at a second examination, the central
corneal epithelium of the left eye was found to be hazy and the
whole bulbar conjuntivæ much congested. “This condition soon
developed into a severe superficial corneitis, which was mainly
limited to a central area of an extent about equal to three-fourths of
the diameter of the cornea, which threatened to slough, a narrow
peripheral ring of the cornea being comparative unaffected. At the
same time the engorgement of the bulbar conjunctiva increased. The
sclera, the iris, and the deeper parts were apparently not involved in
the inflammatory process.”

Disturbances of respiration were observed in a number of cases in

various stages. Cheyne-Stokes breathing was usually a late
symptom. In a case of tubercular meningitis with a tubercular
granulation springing from the left side of the fourth ventricle (Case
82) it was present. Extraordinary slowing of respiration occurred in a
tumor of the right middle cerebellar peduncle and cerebellar
hemisphere which caused irritation and softening of the floor of the
fourth ventricle. The respirations ran as low as four and five per
minute two weeks before death.

Persistent epistaxis and a tendency to hemorrhage from the mucous

membranes were interesting vaso-motor phenomena in a case
situated in the upper left quarter of the pons (Case 84). Profuse
perspiration, more marked on one side, was observed in a case of
tumor in front of the optic chiasm. Polyphagia was observed in two
cases, one a growth of the cerebellum and the other on the floor of
the skull. Polyuria was a very marked symptom in Case 95, a tumor
at the base of the brain at a spot corresponding to the sella turcica,
and diabetes was present in a case of frontal tumor. Albuminuria was
recorded twice—once in the same case in which diabetes was
present, and again in a case of multiple tumor of the supramarginal
convolution of one side and the angular gyrus of the other.
Somnolence was occasionally observed.

Constipation or torpor of the bowels occurs somewhat frequently in

the early stages of the brain tumor, giving place in the terminal
periods to involuntary evacuations. The conditions of the bladder are
practically the same. It is either not involved or suffers from torpor or
paresis of the muscular walls early in the disorder, and later, and
especially very late, incontinence from paralysis of the sphincter
results.

DURATION, COURSE, AND TERMINATION.—The duration of cases of

intracranial tumor is very uncertain. In many of the reported cases no
definite information is given as to the exact length of time from the
initial symptoms until the fatal termination. The few cases in which
the time was recorded showed a duration of from three months to as
many years.

In a few cases, even in some which are not syphilitic in character, a

remission of all the symptoms and what appears to be an
approximate cure sometimes take place, the general symptoms,
such as headache, vertigo, vomiting, spasms, etc., disappearing for
a time. Even the condition of the eyes and the paralysis in rare
instances make marked improvement. In these cases, in all
probability, the progress of the growth of the tumor is arrested either
by the remedies employed or spontaneously, and the acute or
subacute phenomena of congestion, œdema, etc. around the tumor
subside. These patients may remain for a long period or until cut off
by some other disease without any change for the worse; but the
sword constantly hangs above their heads, and any excitement,
traumatism, the abuse of alcohol or other narcotics, an attack of
fever, or some other special exciting cause, may again light up the
intracranial disorder, to then progress more or less rapidly to a fatal
termination.

This fatal termination may occur in various ways. Sometimes a

sudden apoplectic attack occurs. This may be an intercurrent
hemorrhagic apoplexy, although our personal experience would not
lead us to believe this mode of termination is common. In a few
cases the enormous irritation of the cerebral growth suddenly or
gradually inhibits the heart's action through the impression made on
the pneumogastric. Apoplectic attacks which may or may not
terminate fatally sometimes are the result of a sudden giving way of
necrosed brain-tissue, the necrosis having resulted from the
obliteration of numerous blood-vessels by the advancing growth.
Blood-poisoning occasionally takes place from abscesses in
proximity to the tumor. In some cases the patients slowly but surely
emaciate, or are exhausted and worn out by the agonizing pain and
incessant vomiting which they are called upon to endure.
Occasionally a more or less diffused and violent meningitis hastens
the fatal issue.
COMPLICATIONS AND SEQUELÆ.—Tumors of the brain may be
complicated with other affections due to the same cause. Thus, for
example, in a case of gumma other evidences of syphilis may be
present in the form of nodes, eruptions, etc. A sarcoma or carcinoma
of the brain may be associated with similar disease in other organs.
Such affections as cystitis, pyelitis, keratitis, etc., which have been
discussed under Symptomatology, are secondary complications of
cases of tumor. As intracranial tumors almost invariably terminate
fatally, strictly speaking we have no sequelæ.

PATHOLOGY.—We present in tabular form the various classes of

tumors found in the one hundred cases of brain tumor in the table
appended to this article:

Carcinoma 7 Glio-sarcoma 1
Cholesteotoma 1 Gumma 13
Cyst 2 Lipoma 1
Echinococcus 2 Myxo-sarcoma 1
Enchondroma 1 Myxo-glioma 2
Endothelioma 1 Osteoma 2
Fibro-glioma 2 Sarcoma 15
Fibroma 4 Tubercle 13
Glioma 16 Unclassified 16

The histology of tumors of the brain does not in the main differ from
that of the same growths as found in other parts of the body, so that
a detailed description of their structures, even though founded upon
original research, could not offer many novel facts in a field which
has been so thoroughly cultivated. Such a description would
probably repeat facts which have already been presented in other
parts of this work, and which are better and more appropriately put
forth in special treatises devoted to the science of pathology. It is
proper, however, for the sake of convenience and thoroughness, to
make brief mention of the structure of brain tumors, and especially to
dwell upon certain features of these morbid growths which may be
considered characteristic of their encephalic location, and hence
have not only pathological but also clinical interest. It is hardly worth
while to refer to speculations which aim to elucidate the very
foundations of the science, except that in a few of these theories we
gain an additional insight into both the structure and conduct of some
very characteristic brain tumors.

Cohnheim's theory was that tumors are formed from foci of

embryonal tissue which had been non-utilized or left over in the intra-
uterine development of the body. Many have not accepted this idea,
but have rather considered that in tumors we witness a reversion of
tissue to lower or embryonic types.28 Whether we accept either or
neither of these propositions, the idea sought to be conveyed is that
in all these morbid structures we have a tissue of low or degraded
character, springing in most instances from a connective or non-
differentiated tissue. This fact is brought out very clearly in many of
these intracranial growths. Virchow29 has said that tumors originate
in the cells of the connective tissue, although his law has been
condemned as not of sufficient breadth, since it seems to ignore the
epithelial and myomatous tumors. Dermoid cysts, of which an
example is given in the table of spinal tumors,30 are said to illustrate
the embryonic function revived—i.e. the tendency of lower tissues to
spontaneously differentiate into higher and more complex ones.
28 Article “Pathology” in Brit. Encyc., by C. Creighton.

29 Quoted by Cornil and Ranvier.

30 Page 1107.

The gliomata are among the most common and characteristic tumors
of the cerebro-spinal axis, to which system and its prolongation into
the retina they are confined. They invariably spring from the
neuroglia or connective tissue of the nerve-centres, and reproduce
this tissue in an embryonal state. They greatly resemble the brain-
substance to naked-eye inspection, but have, histologically, several
varieties of structure. These variations depend upon the relations of
the cell-elements to the fibres or felted matrix of the neoplasm. In the
hard variety the well-packed fibrous tissue preponderates over the
cell-elements, and we have a tumor resembling not a little the
fibromata (Obernier). The second variety, or soft gliomata, show a
marked increase of cells of varied shapes and sizes, with a rich
vascular supply which allies these growths to the sarcomata. The
elements of gliomata sometimes assume a mucoid character, which
allies them, again, to the myxomata.
 FIG. 43.

Flat Glioma-cell with its Fibrillar Connections (Osler).

FIG. 44.
 (1) Homogeneous translucent fibre-cell; (2) cells like unipolar ganglion-
cells; (3) giant cell (Osler).

W. Osler has recently described31 to the Philadelphia Neurological

Society the structure of certain of these tumors, from which we
abstract the following facts: One point referred to is that gliomata
sometimes contain larger cells and coarser fibres than are usually
shown. The structures are (1) The “spinnen” or spider-cells
(characteristic of glioma), which present variations in size; (2) large
spindle-shaped cells with single large nuclei (some of the largest
cells met with in tumors); (3) cells like the ganglion-cells of nerve-
centres, with large nuclei and one or more processes: some are
balloon-shaped with single processes; they are larger than the
spider-cells; (4) translucent band-like fibres, tapering at each end,
without nucleus or granular protoplasm, regarded as a vitreous or
hyaline transformation of the large spindle-cells. Klebs (quoted by
Osler) holds that the ganglion-like cells are derived from the nerve-
cells of the gray matter, “and that in the development of this variety
all elements of the nerve-tissue participate.” Osler examined the
advancing region of the tumor, and was not able to satisfy himself
that the nerve-cells were in process of proliferation. He thinks they
are connective-tissue elements. He has seen but two out of five
cerebral gliomata which were of small-celled type.
31 “Structure of Certain Gliomas,” Philada. Med. News, Feb. 20, 1886.

The gliomata are subject to fatty degeneration, which usually occurs

in the central (older) portions of the mass. The more vascular forms
are also peculiarly liable to hemorrhage, which is probably caused in
some instances by this process of retrograde metamorphosis. These
hemorrhages resemble apoplexies, not only in their clinical features,
but also on gross examination. Great care is therefore often
necessary at the autopsy to distinguish such a hemorrhage,
occurring as it does in a brain-like neoplasm, from one caused by the
rupture of a diseased artery. The hypertrophy of the pineal gland,
sometimes noted, is caused by the formation of gliomatous tissue.
Under the microscope it is necessary carefully to distinguish some
forms of inflammatory new formations from the gliomata. We have
recently seen, by the courtesy of E. N. Brush of the Pennsylvania
Hospital for the Insane, photographs of microscopic sections from
the ependyma of the lateral ventricles in a case of general paresis,
which showed the structure of this degenerated tissue to be a
compound of fibres and cells of marked resemblance to gliomatous
tissue.32
32 These micro-photographs were prepared in the laboratory of the State Lunatic
Asylum, Utica, New York, by Theodore Deecke.
Sarcomata of the brain are common, as our table shows. In them the
cell-elements predominate, both in the large- and small-celled
variety. They are malignant and grow rapidly. The form known as
alveolar sarcoma, which has a distinct stroma, is to be distinguished
from the cancers; which has probably not always been done.

Tubercle, according to Ross, is the most common of all forms of

brain tumor. Our table shows 13 cases out of 100, the gliomata and
sarcomata being in larger number. Its favorite seat is in the cortex of
both the cerebrum and cerebellum: some observations appear to
show that it is more common in the cerebellum and mid-brain region
than in the fore-brain, and in children than in adults; some of which
points distinguish it from the gummata, which are more common in
adults and occur anywhere. Tubercle is another form of development
from the connective tissues, usually dependent upon a constitutional
taint or predisposition: in it the cell-elements have generally
undergone a degeneration into an amorphous cheesy mass. It is apt
to be multiple and accompanied by a similar deposit in other organs
of the body.

True neuromata are probably very rare growths, and it is likely that
some tumors which have been described as such are really
connective-tissue tumors of a gliomatous nature, in which some of
the cell-elements have been mistaken for the ganglion-cells.
Obernier33 says that these tumors are small and grow from the gray
matter on the surface, also on the ventricular surfaces. They are also
found in the white matter. He says they are only found in persons
having some congenital or acquired aberration; by which is probably
meant some other well-marked neurosis or psychosis. The one
hundred tabulated cases afforded no examples of neuromata.
33 Op. cit.

Myxomata are not, histologically, to be distinguished from the

gliomatous tissues by anything but the peculiar mucoid changes
which their structures have undergone. They are more rare in the
brain, as our tables show, than in the spinal cord.
Lipomata are very rare in the brain, according to most observers.
The table shows but one example. These tumors, as their name
signifies, are made of fat-bearing tissues—another of the connective-
tissue class.

The angiomata, somewhat rarely found within the skull, are noted for
their abnormal development of the vascular tissues: they are
composed mainly of blood-vessels and the connective tissue, which
supports them in closely-packed masses. They also present
cavernous enlargements. They are of especial interest in cerebral
pathology, because the lesion known as pachymeningitis
hæmorrhagica, often found in dementia paralytica, is considered by
some to be angiomatous; although by far the most generally
accepted view of this latter condition is that it is due to arterial
degeneration, and in part is an inflammatory exudate.

Syphilitic tumors, or gummata, are, like tubercle, a special

development with degeneration from the connective tissue, due to a
constitutional taint. This new growth is sometimes single, sometimes
multiple. The corpuscles of the neuroglia are the apparent points of
origin of the tumor, the substance of which is the firm, peculiarly
gummy, and non-juicy material from which the name is derived. It
would be impossible in our allowed space to trace this neoplasm
through the successive stages of its development. It has especial
clinical interest, inasmuch as it and its damage are probably
amenable to specific treatment when it has not progressed to too
great a destruction of brain-tissue.

The true cancers, or epithelial neoplasms, are not a common form of

tumor of either the brain or spinal cord. They present, as in other
parts of the body, a stroma forming alveolar spaces in which are
contained the nests of epithelial cells. These tumors thus present
characteristic differences in their histology from the connective-tissue
or mesoblastic groups, but clinically no very special interest attaches
to them. Their location, the rapidity of their growth, and their fatal
import are points which they share with most other new growths of
the cranial cavity.
The cholesteotomata, or pearl cancers, consist of hardened
epithelial cells which have undergone a sort of fatty degeneration.

The psammomata are loosely described as tumors containing sand-

like bodies, which bodies are normal about the pineal gland. These
sand-like bodies are found in tumors of some histological diversity,
and do not appear to have much identity of their own. They occur in
sarcomata and carcinomata, and are probably not to be
distinguished from mere calcareous infiltration and degeneration.
They are most common in sarcomata, as this is one of the most
common of cerebral tumors.

True osteomata—i.e. tumors with the structure of true bone—are

probably rare in the brain, although more common on the inner table
of the cranium; but the deposition of calcareous salts has been
recorded in a variety of conditions. F. X. Dercum, in a recent paper
read before the Philadelphia Pathological Society,34 has recorded the
autopsy of a paretic dement in which case calcareous deposits were
scattered throughout both hemispheres and the cerebellum. He
believes that “the areas in which the concretions were found were
probably foci of encephalitis of greater intensity than elsewhere. In
these foci inflammatory changes in the walls of the vessels became
pronounced; besides which the vessels increased enormously in
size and number; so marked is this increase that these foci could,
with perfect propriety, be called angiomata.” This is followed by
proliferation of the neuroglia, compression and destruction of nerve-
tissue, and deposit of the calcareous salts especially about and upon
the coats of the vessels. This case illustrates in the simplest manner
the formation of both vascular and sand tumors.
34 The Medical News, April 24, 1886, p. 460.

Pacchionian bodies are very common in the brain, and are really
small fibromata. They may form true tumors (Cornil and Ranvier)
capable of wearing away the bones of the cranium. In fact, even
when small they may have corresponding indentations in the skull.
They are not to be mistaken for tubercle. Clouston35 has described
excrescences from the white matter of the brain, growing through the
convolutions, projecting through the dura mater, and indenting the
inner table of the skull; which new growths he calls hernia of the
brain through the dura. We have not seen such a condition
described elsewhere, and think that we have here probably
Pacchionian bodies growing from the pia mater. They were found in
a case of tumor of the cerebellum.
35 Journ. Ment. Sci., xviii. p. 153.

A cystic formation, constituting a veritable tumor, not unfrequently

occurs in the pituitary body and mounts into the third and lateral
ventricles. Echinococci and hydatids also occur, and have the same
natural history as these parasitic offspring have when found in other
parts of the human body.

Obernier refers to an enchondrosis of the basilar process. Our table

presents one case of enchondroma.

Some of the gross appearances found on autopsies of tumors of the

brain are worthy of note. Often an area of congestion or
inflammation, especially of the membranes, is seen about the new
growth, and the brain-substance in its immediate vicinity is much
more frequently softened. The cerebro-spinal fluid is increased, and,
especially when direct pressure has been exerted upon the veins of
Galen, are found distended lateral ventricles. When a tumor does not
approach the surface, but has attained some size, the hemisphere in
which it is located often has a bulging appearance, crowding over
upon its neighbor, and the convolutions are flattened by the
pressure. The cranial nerve-trunks are occasionally involved in or
stretched by the tumor, and also occasionally the bones of the vault
or base of the cranium are extensively eroded. This happens
especially in cancer and osteo-sarcoma.

A few remarks should be made about the methods of making post-

mortem examinations and the gross appearances and conditions
likely to be found in brain-tumor cases. As not a few intracranial
tumors are connected with the bone or with the dura mater, the latter
being adherent to the skull-cap in some positions because of
inflammation arising from the seat of the growth, especial care
should be taken in removing the calvarium. Examination of the
external surface of the dura mater will sometimes reveal the
presence of a growth beneath or incorporated with this membrane.
The dura mater should not be roughly dragged from the surface of
the brain, but should be carefully removed by a process of partial
dissection. During this process a meningeal growth will sometimes
be found growing apparently from the fused membrane. In such
cases it is usually better to so proceed as not to entirely separate the
outer membrane from the growth. Indeed, this cannot be done
sometimes without injury directly to the specimen, and especially to
its cerebral surroundings. The dura mater having been removed, a
marked opacity, sometimes a dirty-brown hue shading off into a
lighter color, will indicate to the eye the probable presence of a tumor
beneath and growing from the pia mater of the cortex. In such a
case, and even when no such appearance is present, but a tumor is
suspected, the fingers passed carefully over the cerebral surface will
feel a hard, and it may be nodulated, mass at some position. A
growth, having been located in this way, should not be roughly
handled or at once examined by section. An effort should be made to
accurately localize it, not only with reference to lobes, but also with
reference to convolutions and fissures, and even special portions of
these. This is best done, after a thorough examination has been
made of the pia mater, by carefully stripping the pia mater from the
brain, beginning at points some distance from the growth and
gradually approaching it, and leaving the pia mater for a short
distance around the growth connected with it. The location having
been fixed and other portions of the brain having been examined, if it
is not possible or desirable to retain the entire brain as a specimen, a
block should be removed embracing a considerable portion of
healthy brain-tissue on all sides of the tumor. In order to study the
gross internal appearance of the tumor, it is a good plan to make a
clean section through the middle of the tumor. From each side of this
cut fragments can be taken for microscopical examination without
deranging appreciably the size and appearance of the tumor.
When the tumor is not meningeal or cortical, or not situated at the
base or floor of the skull, its presence may be revealed, when it is in
centrum ovale and of considerable size, by either hardness or
fluctuation of the hemisphere in which it is located, this fluctuation
not being due to the tumor itself so much as to the breakdown of
tissue around it. Large sections in known positions with reference to
convolutions and ganglia should be made when examined for tumors
deeply situated. If possible, sections close to and just before and
behind the growth should be made, so as to assist in the accurate
localization.

Small tumors are not infrequently overlooked by careless observers,

and even growths of considerable size have escaped discovery by
one examiner to be found by another. Tumors in certain special
localities, as between the temporo-occipital lobe and the superior
surface of the cerebellum in the great longitudinal fissure, or small
growths in the substance of the cerebellum or deep in the Sylvian
fissure, are more likely than others to be passed by, although this, of
course, is not likely to occur when the examination is made by a
competent or careful physician.

DIAGNOSIS.—The diagnosis of the existence of an intracranial tumor,

as a rule, is not difficult. It can be made with greater certainty than
that of almost any other serious encephalic disease.

It is sometimes important to decide as to the nature of an intracranial

neoplasm, particularly whether or not it is syphilitic. Little is to be
gained by following the plan adopted by some physicians, of treating
all cases as if they were due to syphilis, on the principle that these
are the only forms of tumor which can be reached by treatment. The
pitiable condition of such patients is sometimes thus made worse. In
every case careful and persistent efforts should be made to obtain
an authentic previous history from the patient. Whenever possible
the physician should search directly for the physical evidences of the
former existence of syphilis—for cicatrices on the genitals and
elsewhere, for nodes and depressions, for post-cervical and other
swellings, etc. A history of previous disease of the throat and of
pains in bones and nerves, of epileptiform attacks, of headache, and
eye symptoms which have disappeared under treatment, should be
sought out. It is not well to give too much credence to the stories of
patients, who are not always willing to admit their past lapses from
virtue; but, on the other hand, the plan of suspecting everybody who
presents advanced cerebral symptoms is often a grievous wrong.
Not infrequently external cranial nodes are present in cases of
intracranial syphilis.

Carcinomata and sarcomata, particularly the former, are

comparatively rapid in their progress. They sometimes involve the
bones of the skull, even to the extent of perforation.

The existence of an inherited tendency and of tuberculosis in other

organs, with the special phenomena of general tuberculosis, assists
in the diagnosis of tubercular tumors.

The frequent occurrence of gliomata in early life, and the

comparatively frequent absence of severe irritative symptoms, with
the well-preserved general nutrition of the patient, speak for these
growths.

Cerebral abscess is, on the whole, more difficult to diagnosticate

from intracranial tumor than any other affection. Abscess, however,
more frequently than tumor, can be traced directly to a traumatism. It
is often associated with disease of the internal ear. Obernier speaks
of the headache of cerebral abscess as slight, but this does not
correspond with usual experience. Headache, on the whole, may be
oftener absent or less agonizing in abscess than in tumor, but it is
frequently present, and sometimes of great severity. Its greater
mildness in a few cases is to be explained by the fact that abscess
does not produce so much pressure within the intracranial cavity,
and does not so frequently cause irritation of the branches of the
trigeminus in the dura. Undoubtedly, the symptoms of abscess often
remain for a long time comparatively latent, with then a sudden
outburst of violent symptoms. The course of brain tumor is more
uniformly and steadily progressive, and febrile phenomena, the
results of pyæmia, are of more frequent occurrence in abscess than
in tumor.

In old cases of tumor it is sometimes necessary to differentiate

between it and the results of various forms of apoplexy, such as
hemorrhage, thrombosis, and embolism. Cerebral hemorrhage,
embolism, or thrombosis leaves a condition of paralysis, sometimes
with, but usually without, accompanying spasm or convulsion, which
simulates closely the paralysis and other permanent conditions of
cases of tumor occurring in the same cerebral locality. In these
cases, in the first place, the history of the disease will throw
considerable light upon the diagnosis. In both hemorrhage and
embolism the history is usually one of a sudden attack without
special premonitory symptoms. Hemorrhage gives usually a
precedent history of diseased kidneys, hypertrophied heart, or
atheromatous blood-vessels, and occurs generally in advanced life;
embolism, a history of rheumatism and valvular disease of the heart,
occurring at any period of life, early or late. In brain tumor the
previous history is usually one of traumatism, of constitutional
infection, or of a special predisposing diathesis. Blows and falls upon
the head are common antecedents, or a history of syphilis,
tuberculosis, scrofula, or cancer is present. Tumor, like embolism
and unlike hemorrhage, may occur at any time of life. While slight or
dull headache, with more or less vertigo, may be present in cases of
hemorrhage and thrombosis, the severe and often agonizing
headache, with vomiting and serious vertiginous attacks, which
precedes the paralytic or other phenomena of tumor, is a much more
conclusive symptom in the latter cases than in the former. Choked
discs and optic neuritis are much more likely to occur in tumor than
in the other affections.

Brain tumor must sometimes be diagnosticated from the head

symptoms of some form of Bright's disease. A case not long since
presented itself to one of us with a history of having suffered at
frequent intervals for two years with headache of gradually
increasing severity. Dimness of vision and slight temporary œdema
of the feet, circumscribed and painful swellings along the lymphatics
of the thighs and legs, with some mental irritability, were other
marked symptoms. The patient had been attended by several
physicians of prominence, one of whom had diagnosticated tumor of
the brain. The violent, apparently agonizing headache, with the
diminution of vision, and the absence of marked symptoms indicating
other organic disease, made the diagnosis of a growth in some non-
excitable region of the cerebrum most probable. Examination of the
urine showed no albumen. Careful examination of the eye-ground
with the ophthalmoscope, however, revealed the appearances of
retinitis albuminurica. Under a treatment directed to the relief of
chronic nephritis the patient's headache and other symptoms
improved.

It must not be forgotten just here, however, that, on the one hand,
ophthalmoscopic appearances very similar to those of albuminuric
retinitis are sometimes present in rare cases of brain tumor, and also
in other constitutional disorders, such as leukæmia; and, on the
other hand, that, as stated by Norris,36 exceptional forms of
albuminuric retinitis have been reported where the only change seen
in the fundus oculi was pronounced choking of the disc.
36 Op. cit.

Intracranial tumors must be diagnosticated from meningitis in its

various forms. In children tubercular meningitis sometimes closely
simulates brain tumor. Tumors of the brain are comparatively rare in
children, but, as has already been shown, gliomata and other tumors
do sometimes occur in early life. The course of tubercular meningitis,
whether in children or in adults, differs from that of brain tumor. It is
more irregular in its method of advance, or if it shows the regularity
which is sometimes present, and which has led authors to subdivide
it into three more or less completely separable stages, the symptoms
of these stages do not correspond with any closeness to those of the
initial, middle, and terminal periods of brain tumor, as already given.
Headache is usually present in both affections, although the absence
of headache in some cases of gliomata in children must be here
borne in mind. When headache is present in tubercular meningitis, it