US2860985A - Stabilization of photographic silver halide emulsions and means of increasing contrast - Google Patents

Description

United StategP O This inventionrelates .to' fog-inhibiting agents and stabilizers for photographic emulsions, and to photographic emulsions containing them. 7

It is well known that photographic emulsions on storage .tend to lose sensitivity and to become spontaneously developable without exposure to light. There is'norm'ally a detectable amount of the silver salt reduced during development in the areas where no exposure was given; this is commonly called fog, and sometimes called chemical fog where it is necessary to distinguish between it and the effects of accidental. exposure tora'diation; in this invention, we are not concerned with the latter.- i i Fog, depends both on the emulsion and the conditions of development; for a given emulsion it increases with-the degree of development. With constant development conditions, it tends to increase with time, temperature and relative humidityof storage conditions; it iscommon practice to make accelerated tests ofv the stability ofp'hot'o- SILVER gr'aphicemulsions by storage at increased temperature 1 "or humidity, or both. It is, of course, desirable to-h'ave emulsions, as stable as possible under the conditions of high temperature and humidity which may occur" in tropical climates-for example. Fog usually appears ov'er the whole area of the sensitive coating, but when severe, it frequently is non-uniform. Fog may al'sobe caused by exposure to chemicals, for example, hydrogen sulfide and other reactive sulfur compounds, hydrogen" peroxide vapor, and strongly reducing materials. While v antiice benzothiazoline nucleus (wherein the benzene ring thereof can contain substituents, such as chlorine, bromine, methyl, ethyl, methoxyl, ethoxyl, etc.), and Q represents an oxygen atom or imino (NH). Especially useful compounds embraced by Formula 1'' above are those represented by the following general formula:

II. I

wherein R, D and Q each have the values given above, 11 is 1 or 2 and R represents a hydrogen atom or a carboxyl group (i. e., carbox-yl" or a water-soluble salt thereof, such as the alkali metal salts (e. g., sodium, potassium,

foggants and stabilizers may protect, tosome extent,

against such elfects, it is normally understood that. an a'ntif'o'ggant protects .against spontaneous growth 'of fog during prolonged storage or storage at high: temperatures and humidities, or during development to max'imum contra'st and'speed', or both.

-It is, accordingly, an object of our a method for stabilizing photographic emulsions. .A further object of our invention is to maintain the sensitivity and fog of silver halide emulsions at or close to initial optimum values under keeping conditions of high temperature and humidity. A further object is to provide photographic silver halide emulsions containing antifoggants' or stabilizers. Other objects will become apparent from a consideration of the following description and examples.

The above objects are accomplished byadding to the photographic silver halide emulsion a compound selected from those represented by the following general formula:

I. s R I a l I C. II Q invention' to' provide sents the non-metallic atoms necessary to complete a thia'z'olidine nucleus, a tetrahydrothiazine nucleus or a etc.), or ammonium or organic amine salts, such as triethylamine, triethanolamine, pyridine, etc.).

We have also found that the compounds represented by FormulasI and-II above have the useful. property of increasing contrast in various types of emulsions, such as ordinary silver chloride emulsions which are used in preparing photographic. prints. The fog-inhibiting and stabilizing properties of these compounds are especially outstanding, and for this reason, these compounds are referred to as fog inhibitors below, although it should be understood that they have other useful. properties as well.

The fog inhibitors which we propose to use are, added to the emulsion duri g; thepro'c'ess of manufacture, to avoid loss of sensitivity and to inhibit the growth of fog with passage of time under non-ideal conditions of storage.

A solution of the compounds of the invention when added in suitable concentration," before coating, to unsensitized, chemically sensitized, or optically sensitized photographic emulsions does not appreciably affect the sensitornetric values for sensitivity and fog when measurementsare made soon after coating. When sen'sitometric measurements are made at appreciable intervals of time, at elevated temperatures and dry or somewhat humid conditions these compounds do stabilize photographic speed and maintain fog at a low level.

The preparation of silver halide emulsions involves three separate operations: (1) the emulsification and digestion or ripening of the silver halide, (2) the freeing of the emulsion from excess soluble salts, .usually by washing, and (3) the second digestion or afterripening to obtain increased sensitivity (Mees: The Theory of the Photographic Process (1942)). We prefer to add the fog-inhibiting agents after the final digestion or afterripening, although they can advantageously be added prior to digestion.

Listed below are a number of compounds coming within the scope of the above general formulas, which we have found to be particularly advantageous inpracticing our invention.

Lactam of 4carboxy-2-(o-carboxyphenyl) thiazolidine H 5-1mino1soindolo [1,2-b] thiazolidine-3-carboxy1ic acid Lactam of Z-(B-carboxyethyl)-2-methylthiazolidine Lactam of 2-(o-carboxypheny1)thiazolidine V S\ /H Hac C--- HOOG H i 6-ket0-2,3,6,10b-tetra hydroisoindolo [1,2-b] thiazineicarboxylic acid 11-keto-4b,1l-dlhydroisoindolo[1,2-b1benzothiazole H 11-imino-4b,11-d1hydroisoindolo [1,2-b]benzoth1azole The photographic emulsions used in practicing our invention are generally of the developing-out type; also, it is to be understood that photographic emulsions of varying halide content can advantageously be used. The antifoggant compounds used in our invention have been found particularly useful when employed in conjunction with gelatino-silver-bromiodide emulsions, although they can also be advantageously employed for stabilizing other silver halide emulsions, such as gelatino-silver-chloride, bromide, chlorobromide, chlorobromiodide, etc.

Patent 2,448,060, and as antifoggants in higher amounts, as described in Trivelli and Smith U. S. Patents 2,566,245 and 2,566,263.

The emulsions can also be chemically sensitized with gold salts as described in Waller and Dodd U. S. Patent 2,399,083, or stabilized with gold salts as described in Damschroder U. S. Patent 2,597,856 and Yutzy and Leermakers U. S. Patent 2,597,915. Suitable compounds are potassium chloroaurite, potassium aurithiocyanoate, potassium chloroaurate, auric trichloride and 2-aurosulfobenzothiazole methochloride.

The emulsions can also be chemically sensitized with reducing agents such as stannous salts (Carroll U. S. Patent 2,487,850), polyamines such as diethylene tri amine (Lowe and Jones U. S. Patent 2,518,698), polyamines, such as spermine (Lowe and Allen U. S. Patent 2,521,925), or bis-(fi-aminoethyl) sulfide and its watersoluble salts (Lowe and Jones U. S. Patent 2,521,926).

The emulsions can also be stabilized with the mercury compounds of Allen, Byers and Murray U. S. Patent 2,728,663, Carroll and Murray U. S. Patent 2,728,664, and Leubner and Murray U. S. Patent 2,728,665.

The stabilizing and contrast-increasing compounds of our invention are effective in the presence or absence of optical or spectral sensitizing dyes. Since spectral sensitizing may affect stability of emulsions with respect to sensitivity, fog and latent image changes, the action of the compounds of our invention is not completely independent of spectral sensitizing or other emulsion variables. We have found, however, that both unsensitized emulsions and emulsions sensitized with cyanine or mero- -cyanine dyes, or both, can be treated with the aboveformulated compounds according to our invention. We have noted in certain instances that the compounds of our invention have a sensitizing action of their own in additio to the other effects noted above. In other instances, these compounds stabilize the speed of the photographic emulsions, when the emulsions are subjected to adverse keeping conditions.

The following table will serve to illustrate the antifoggant and stabilizing action provided by the compounds of our invention, when added to an ordinary negative gelatino-silver-bromiodide emulsion. The antifoggant and stabilizing action was determined by incubation of the emulsions for one week at a temperature of 120 F., and two weeks at a temperature of 120 F. in certain cases, and at constant relative humidity (obtained by placing the emulsions in closed containers, the embient temperature being about 70 F. and relative humidity about prior to sealing the containers).

The etficiency of the various antifoggants was determined by measuring the speed, gamma and fog of the in- The emulsions can also be chemically sensitized by any i of the accepted procedures.- The emulsions can be digested with naturally active gelatin, or sulfur compounds can be added such as those described in Sheppard U. S. Patent 1,574,944 and U. S. 1,623,499, and Sheppard and Brigham U. S. Patent 2,410,689. Y i i The emulsions can also be treated with salts of the noble metals such as ruthenium, rhodium, palladium, iridium and platinum, all of which belong to group VIII of the periodic table of elements and have an atomic weight greater than 100. Representative compounds are ammonium chloropalladate, potassium chloroplatinate and sodium chloropalladite, which are used for sensitizcubated emulsions containing an antifoggant and comparing these measurements with those of the same batch of emulsion before incubation, as well as comparing these measurements with those made on an identical batch of photographic emulsion containing no antifoggant.

The tests were made using high-speed, negative-type silver 'bromiodide emulsions coated on cellulose acetate supports. After drying the coatings, they were exposed on an Eastman sensitometer Type Ib for second with a -watt sunlamp. The coatings of Examples 1 to 4 were then developed for 4 minutes in Developer D-l9 having the following composition:

Water to make one liter.

The coatings of Examples 5 and 6 were developed for '5- minutes in Developer DK-SO having the following composition:

The speed, gamma and fog for each of the emulsion coatmgs were measured as indicated above. In Examples 1 to 4, the speed figure is given as the conventional 30/13 speed, whereas the speed-figure given in Examples 5 and 6 is a comparative speed obtained by the, formula: 100(1-LogE), where E is the exposure in meter-candle-seconds of sunlight quantity required to produce a density of 0.30 above fog. A difference of 30 pointson this :speed measurement represents a change of 100% in speed. The results obtained were as follows:

acid or alkaline photographic ilver halide emulsions. In alkaline emulsions, water s'oluble salts of the compounds of Formula I or II above which contain a carboXyl group are formed to some extent, If desired, these compounds can be first addedtoa dilute alkaline solution and this solution added directly to the photo'- graphic silver halide emulsion;

Many of the compounds useful in our invention are novel. The compounds of Formulas I and II wherein R has the values given above, Q represents an oxygen atom, and D represents the non-metallic atoms necessary to completea thiazolidine nucleus or a tetrahydrothiazine nucleus can be prepared by heating together a compound selected from those represented by the following general formula:'

TABLE A Compound Fresh Test Incubation, 1 week at'120 F. Incubation, 2 weeks at 120 F. Example (g./mol. i s r 1 a a 30/E Speed Gamma Fog 30/E Speed Gamma Fog 30/E Speed Gamma Fog None 2,800 1.14 0.10 3,150 0.97, 0.28 2 (0.03) 3,100 1.14 0.09 4,350 1.00 0.19 ,None 3, 450 1.27 0.10 4, 350 1.08 0.30 3 (0.1) 3, 500 1. 13 0.09 5,000 1. 17 0. 17 None 2, 750 1. 30 0. 11 51, 150 1. 17 0. 4 (0.03) 3, 450 1.22 0.08 5,600 1.05 0.13 None 2, 750 1. 0.11 4, 150 1. 17 0.25 l (0.03) 3, 450 1.22 I 0.08 5,600 1.05 0.13

GB Speed OR Speed None 318 1. 36 0. 10 317 1. 13 0. 16 296 0.90 7 (0.005) 318 1.49 0.10 321 1.18 0.12 303 1.01 0.28 None 318 1 36 0.10 317 1.13 0.16 296' 0.90 0. 40 8 (0.005) 313 1.44 0.12 320 1.15 0.13 305 1.06 0.26

In a manner similar to that illustrated in the above e-Xamples, other compounds selected from" those repr'e sented by the above general formulas (I andII) can be incorporated in photographic emulsions for the purpose of stabilization. The fog-inhibiting agents useful in practicing our invention'can be used in various kinds of photographic emulsions. In addition to being useful in ordinary non-sensitized emulsions, they can also be used in orthochromatic, panchromatic and X-ray emulsions. If used with sensitizing dyes, they can be added to the emulsion before or after the dyes are added. Suitable dispersing agents for the silver halide used in our invention comprise gelatin or other colloids, such as colloidion, albumen, cellulose organic derivatives, synthetic resins, etc. Synthetic dispersing agents are particularly useful where it is desired to demonstrate the effect of the compounds of our invention in increasing the contrast or gamma, in silver chloride emulsions. The compounds can also be used in emulsions containing incorporated couplers or emulsions which are developed in processing solutions containing couplers.

The optimum amount of compound to be used as a fog-inhibiting agent can be determined by making the customary tests employed in emulsion making. Of course, the optimum amount for a given emulsion will vary depending upon the presence of emulsion addenda, such as chemical sensitizers, spectral sensitizing dyes, etc. In general, we have found that from about 0.002 to 3.0 g. of compound per mole of silver halide is sufiicient to-provide an optimum amount of stabilization without serious desensitization.

Instead of adding the fog-inhibiting agent directly to the photographic emulsion, it is sometimes desirable to incorporate the fog-inhibiting agent in a separate layer.

which is placed in contact with the silver halide emulsion layer which is to be stabilized. Under such conditions, of course, it is advisable to use a higher concentration of fog-inhibiting agent than indicated above. The antifoggants of our invention function advantageously in wherein R represents a hydrogen atom or a carboxyl group, and r1 rep'rfesents'l or 2 witna ketone compound selected from those represented by the following general formula:

wherein R has the values given above and R represents a ,fi-carboxyethyl group or an o-carboxyphenyl group (substituted or not, as indicated above). Advantageously, the compounds of Formulas I and II wherein R represents a methyl group are prepared by reacting an acid-addition salt of a'compound of Formula III '(e. g., the hydrochloride, hydrobromide, etc.) with the ketone compounds of Formula IV wherein R is methyl. In such cases, R is generally a carboxyl group and R is a fi-carboxyethyl group. These condensations can .be accelerated through the use of an acid-binding agent, such as pyridine, quinoline, triethylamine, etc., to remove the acid liberated from the acid-addition salt of the compound of Formula III.

The compounds of Formula I wherein D represents the non-metallic atoms necessary to complete a benz'o thiazole nucleus can be prepared by reacting an o-a'minofbenzenethiol with the compound of Formula IV; This reaction is quite similar to that wherein the compounds of Formula III (R is hydrogen) are reacted with the compounds of Formula IV.

The compounds of Formulas I and II wherein Q is imino, D is o-phenylene and R is hydrogen can be'prepared by condensing o-c yonabenzaldehyde' with the, compounds of Formula 111. These reactions can be illustrated by the following examples.

Example A.Lactam of 4-carboxy-2-(fl carboxyethyl)- Z-methylthiazolidine cm-sH The resulting solution was refluxed for ten minutes and allowed to cool. The product separated as an oil. The mixture was diluted with water and the product extracted B l I with ether several times. On evaporation of the ether HNHHC1 22 extract (dried over anhydrous magnesium sulfate) the OOH CH; product crystallized out. The product recrystallized from 4 alcohol-water melted at 97 100 C.

Exan zple E.-5-imin0isoindol0[1,2-b]thiaz0lidine-3-car- I 0 boxylic acid HCN CH2 H20 l A solution of 7.8 g. (.05 mole) cysteine hydrochloride 000]; 0*CH dissolved in 20 ml. hot water was added to a solution of o-cyanobenzaldehyde in 20 cc. hot alcohol. On addi- A mixture of 7.85 g. (0.05 mole) of cysteine hydrotion of 4 ml. of pyridine the hot solution began to reflux chloride and 5.8 g. (0.05 mole) of freshly distilled levusp n y- After fl g for two mmutes, the l linic acid in 50 ml. of ethyl alcohol was treated on the t1011 Was allowed to C001 and filteredfifystalllllfi steam bath for 5 mi ute nd 4.1 1, (0,05 l f product was recrystallized from alcohol witha little water pyridine were then added. The reaction mixture was j- DeCOmPOSCd 9 1 The heated on the steam bath for 5 e i ut 50 1, f evidencefor the ring closed amidlne structure was estabethyl alcohol were added and the mixture was allowed lished 011 t basis of q 0f tl'lple'bolld adsorptlon to cool. A small amount of cysteine was filtered off, and 111 i I'eg10I1 Q l 111 l Infrared and the 6X33- the clear mother liquor upon standing for several weeks Q y f fy 0f the llltfogen as Shown y the p at C. gave long colorless needles (2.5 grams in the tr16 tltratlon curve.

first crop) hich lted at 198 199 C, ft d i 25 In a similar manner, other compounds coming within t 60 C, und vacuum Formulas I and II above were prepared. The following table summarizes the method of preparation and the phy- Example B' Lactam iffi' sical constants of a number of compounds embraced by t me these formulas, including the compounds of Examples A A solution of 7.7 g. (0.10 mole) of fl-mercaptoethylto E above.

' TABLE B Analysis Calcd Analysis Found Com- Thiol Reactant Formula IV Reactant M. P 0. pound O H N O H N cystelueJIC levulinic a 198-9 47. 7 5. 47 6. 97 48. 2 5. 9 7.3 do phthalaldehydic acid 218-20 56.2 3.83 5.95 56.7 4.10 6.3 o-cyunobenzaldehyde- 166 56.4 4.27 11.96 55.9 4.4 11.8 levulinic acid... B.P. 8)6 (2 53.5 7.0 8.9 53.6 6.9 8.2

[11111. 5 do phthalaldehydic acid. 97-100 62.8 4.7 7.3 62.7 4.8 7.2 6.-" homocysteine.HCl do 232-3 57.8 4.4 57.9 4.0 7. o-aminobenzenethiol do 172-4 70.3 3.8 70.2 3.0 s do o-eyanobenzaldehyde- 192-3 dee 61.2 4.0 10.2 60.9 4.3 10.1

amine in 25 ml. water was mixed with 25 ml. of an alcoholic solution of levulinic acid (12.8 g., 0.11 mole).

Example C.Lactam of 4-carboxy-2-(o-carboxyphenyl)- thiazolidine To a solution of 3.0 g. (0.02 mole) of phthalaldehydic acid in 60 ml. of hot ethanol were added 3.14 g. (0.02 mole) of cysteine hydrochloride in 20 ml. of hot water. After adding 1.58 cc. (0.02 mole) of pyridine, the reaction mixture was warmed on the steam bath for 15 minutes, and left at F. for two days. The solution was then heated on the steam bath and 75 ml. of water were added. Upon cooling, white needles were formed melting at 161l62 C. When this material was dried at C. under vacuum, the melting point was raised to 218220 C.

Example D.- Lac!am of 2-(o-carboxyphenyl)thiazolidine A solution of 16.5 g. phthalaldehydic acid (0.11 mole) in 25 ml. ethyl alcohol wasadded to 25 ml. of an aqueous solution of 7.7 g. (0.10 mole) fi-mercaptoethylamine.

As indicated above, the compounds represented by Formula I (or II) can be used to increase the contrast and raise the maximum density of certain photographic silver halide emulsions. These compounds also have the effect of shortening the toe portion of the characteristic curve. The efiect of these compounds is illustrated below with respect to a fine-grained pure silver chloride emulsion. This emulsion was prepared using a gelatin derivative, as described in U. S. Patents 2,614,928 and 2,614,931. The emulsion was unripened and unsensitized (chemically or optically) and inert gelatin was used as the vehicle.

The silver chloride emulsion described above was divided into separate portions and certain of these portions treated with one of the compounds embraced by Formula I or II above, in the amounts indicated in the following table. The separate batches of emulsion were then coated on a clear film base and given a 30-second exposure with a 500-watt lamp, 3000 K., in an Eastman Type Ib sensitometer. The coatings were then developed for 4 minutes in Eastman Kodak Developer D-1. The speed, gamma and fog for each of the coatings were then vmeasured in the manner shown in Examples 5 and 6 of Table A above. The maximum density of the coatings was also measured. The concentration of the compound of Formula I or II is given on a molar equivalent basis equal to l g. of 4 carboxythiazolidine per mole of silver halide. The results obtained follow:

- 'Qompounds similar to those zof Formula labove'wherein D represents the non-metallic atoms necessary :toTc'omplete 1a heterocyclic nucleus containing .6' atoms. in the heterocyclic ring, such as tetrahydrothiazine, can also be used to treat photographic silver halide emulsions. Such compounds do not, however, generallyprovide the useful results shown in the above tables.

The optimum amount of compound to be used as a contrast-increasing agent can "be determined by making the customary tests employed in emulsion making. As indicated above with respect to the fog-inhibiting action of these compounds, the optimum amount will Vary depending upon the presence ,of various emulsion addenda. In general, it has been found that from about .1002 g. to 3.0 g. of compound per'mol. of silver halide is sufficient to provide an optimum amount of contrast increase without serious desensitization. v f

"What we claim ;as our invention and desire secured by Letters Patent of the United States is: i 7

1. A photographic silver halide emulsion containing a compound selected from those represented by the following general formula:

8 'R 131 \CD [l 'Q whereimR represents a member selected from the group consisting of a hydrogen atom and a methyl group, D represents a member selected from the group consisting of an ethylene group and an o-phenylene group, D; represents the non-metallic atoms necessary to complete a heterocyclic nucleus selected from the group consisting of a thiazolidine nucleus, a tetrahydrothiazine nucleus, and a benzothiazoline nucleus,;and Q represents a member selected from the group consisting of an oxygen atom and an imino group.

2. A photographic silver halide emulsion as defined by claim 1 wherein the silver halide is silver bromiodide. 5. Aphotographic silver halide emulsion as defined by claim 1 wherein the silver halide is silver chloride.

4. A photographic silver halide emulsion containing a compound selected from those represented bythe following general formula:

wherein R represents a member selected from the group consisting of a hydrogen atom and a methyl group, R represents a member selected from the group consisting of a hydrogen atom, a carboxyl group, and a watersoluble salt of a carboxyl group, D represents a member selected from the group consisting of an ethylene group and an o-phenylene group, n represnts a positive integer of from 1 to 2, and Q represents a member selected from the group consisting of an oxygen atom and an imino group.

nrh ps i hiasilxer ing a compound selected from those represented following general formula:

w i R' p e e t a m be s eqted' m therer a consisting of a hydrogen atom and a-rnethyl ,group k represents a member selected from the group consisting of a hydrogen atom, a carboxyl group, and a water-soluble salt of a carboxylgroup, D represents a member selected from the group consisting of an ethylene group and an o-phenylene group, n represents a positive integer of from 1 to 2, and Q represents a member selected from the group consisting of an oxygen atom and an imino no n 1 .6. A photographic :siluer chloride ;emulsion;containing a compound selected from those represented by-ythe" fol- :lowing;general.=formula:..

wherein R represents a jmember selected from the group consisting of a hydrogen atom and a methyl group, R represents amernber selected from the-group consisting of alayclreg a a em. 1 c h x roup-ma u e -so uble in, D

wherein D represents an ethylene group and D represents the non-metallic atoms necessary to complete a thiazolidine nucleus.

.8. A photographic gelatinoesilver-halide emulsion containing a compound represented by :the .following'formula:

9. A photographic gelatino-silver-halide emulsion containing a compound represented by the following general formula:

wherein D represents an o-phenylene group and D represents the non-metallic atoms necessary to complete a thiazolidine nucleus.

10. A photographic gelatino-silver-halide emulsion bromiodide emulsion gcontain containing the compound represented by the following formula:

Hood

11. A photographic gelatino-silver-halide emulsion containing a compound selected from those represented by the following general formula:

13. A photographic gelatino-silver-halide emulsion containing the compound represented by the following formula:

14. A photographic gelatino-silver-halide emulsion containing a compound selected from those represented by the following general formula:

L D a qlI wherein D represents an o-phenylene group and D represents the non-metallic atoms necessary to complete a benzothiazoline nucleus.

15. A photographic gelatino-silver-halide emulsion containing the compound represented by the following formula:

s /H c l 12 16. A photographic gelatino-silver-bromiodide emulsion containing a compound selected from those represented by the following general formula:

wherein D represents an ethylene group and D represents the non-metallic atoms necessary to complete a thiazolidine nucleus.

17. A photographic gelatino-silver-bromiodide emulsion containing a compound represented by the following general formula:

wherein D represents an o-phenylene group and D represents the non-metallic atoms necessary to complete a thiazolidine nucleus.

18. A photographic gelatino-silver-bromiodide emulsion containing a compound selected from those represented by the following general formula:

t Ila wherein D represents an o-phenylene group and D represents the non-metallic atoms necessary to complete a thiazolidine nucleus.

19. A photographic gelatino-silver-bromiodide emulsion containing a compound selected from those represented by the following general formula:

-\ IlI\ F Ila wherein D represents an o-phenylene group and D represents the non-metallic atoms necessary to complete a benzothiazoline nucleus.

References Cited in the file of this patent UNITED STATES PATENTS 2,566,658 Fry Sept. 4, 1951 2,683,709 Dickey et al July 13, 1954 2,728,667 Knott et a1. Dec. 27, 1955 2,790,791 Towne et al Apr. 30, 1957

Claims (1)

1. A PHOTOGRAPHIC SILVER HALIDE EMULSION CONTAINING A COMPOUND SELECTED FROM THOSE REPRESENTED BY THE FOLLOWING GENERAL FORMULA: