Spectrophotometric Determination of Fluorine in Rocks RICHARD
P.
HOLLINGWORTH
Department o f Geology, Universify o f Durham, Durham, England This work was initiated in order to develop a simple and accurate method for determining fluorine in metamorphic rocks. A sodium peroxide decomposition, followed b y a precipitation of silica and alumina and a steam distillation of the filtrates, gave quantitative isolation of fluorine. Aliquots of distillate were titrated spectrophotometrically with thorium in the presence of sodium 2-(p-sulfophenylazo) 1,8 dihydroxynaphthalene- 3.6disulfonate(SPADNS). The method was tested with synthetic rocks and soda feldspar, to which known amounts of fluorine were added. Aliquots were titrated spectrophotometrically to within 1 y of fluorine. Results with G-1 and W-1 are compared with analyses reported b y others. Replicate results for ten metamorphic rocks indicate a maximum proportional deviation of 2.OY0.
-
U
-
methods for the determination of fluorine in rocks require a twostage isolation. The first stage involves removal of silica, alumina, or both, commonly by the Berzelius method with an improvement introduced by Hoffman and Lundell (9). Grimaldi, Ingram, and Cuttitta (6) have described a zinc oxide-sodium carbonate fusion with direct distillation of filtrates in a phosphoric-perchloric acid medium. Shell and Craig (18) have reported a single precipitation-filtration removal of silica and alumina by a zinc oxidesodium carbonate fusion with addition of ammoniacal zinc carbonate to the water leach. Chu and Schafer ( 5 ) have used a sodium peroxide decomposition for the determination of fluorine in catalysts containing silica and alumina. Their peroxide decomposition has three advantages for silicate rocks: Decomposition is complete ( I C ) , m t e r solubility of the frit eliminates care over ieachings, and a simple and rapid precipitation gives nearly complete removal of silica and alumina in a single step. I n the second stage of isolation fluorine is recovered from the filtrates by some modification of the Willard and Winter distillation (21). A constant temperature steam distillation still designed by Huckaby, Welch and hletler (10) was chosen for its semiautomatic operation. Colorimetric methods for the determination of microgram amounts of fluoSUAL
1 130
ANALYTICAL CHEMISTRY
rine were surveyed. A spectrophotometric adaptation of the Steiger-Merwin peroxidized titanium method, as described by Groves ( 7 ) ,has been reported by hIonnier, Waucher, and Wenger (14). With the instrument in this laboratory, fluorine down to 5 to 7 y can be determined to within 3 to 4 y . A spectrophotometric adaptation of the sodium alizarin sulfonate-thorium method has been described by Icken and Blank (11). Their results suggest that fluorine in the range 0 to 50 7 can be determined to within 1 y. However, the method requires freshly prepared color reagent and repeated use of standard solutions with the assays. Banerjee (2) described a colorimetric determination of pure solutions of alkali fluoride with a new dye, SPADNS, which is the sodium salt of 2-(p-sulfo. phenylazo) - 1,s-dihydroxynaphthalene3,6-disulfonic acid. This dye forms a blue-violet lake with thorium ions; in the presence of alkali fluoride thorium tetrafluoride forms preferentially, the dye keeping its crimson color. According to Ranerjee ( 2 ) this thoriumSPADNS-fluorine system has three advantages: The color change is marked from crimson to blue-violet a t the end point, the reactions of thorium and fluoride in the presence of the dye are immediate, and the temperature coefficient is low. lloreover, the high titers required for titration of fluoride a t p H 2.9 to 3.1 render possible semimicrotitration of microgram amounts of fluoride.
entation in the light path for each dctermination. The cover and top of the cell are painted black, and the side surfaces are roughened with fine sandpaper: Unicam SP. 600 spectrophotometer for the titration. The absorbance is checked at 580 mp where the difference in absorbance for the thorium lake and the dye is greatest. Sodium peroxide. Zinc sulfate solution. Dissolve 36 grams of zinc sulfate heptahydrate in 100 ml. of distilled water. Indicator. Dissolve 0.5 gram of malachite green oxalate in 50 ml. of water. Sulfuric acid, 1 to 3, 20% sodium hydroxide. Sodium sulfate solution, l%, made alkaline to phenolphthalein. Concentrated sulfuric acid.
58DF
Figure 1. APPARATUS AND REAGENTS
The constant temperature steam distillation still designed by Huckaby. Welch, and Xetler (10) for isolation of fluorine was modified by use of a Davies double-surface condenser for collection o distillate and made of smaller dimension. Cnnibridge pH meter. A 4-em. light path cuvette of total volunie 45 ml. and cuvette holder made of Perspev (Lucitc) !\as used for the spectrophotoinetric titration. I t s construction is illustrated in Figure 1. A block mounted on top of the cover for the cell compartment has a flat surface, AB, which $ts flush with a similar surface, A'B , on the cell, when the latter is in the cover hole. This arrangement ensures exact positioning of the cell in the light path. The cell is marked so that it has the same ori-
A\:
Twv-
",?*li 3r
CELL
Titration cell for Unicam
SP. 600 spectrophotometer Treated Soda - Lime Glass. Place about 250 grams of glass powder in 200 nil. of 1 to 2 sulfuric acid. Heat the mixture a t 70" C. for a n hour. Cool, filter, wash nith boiling distilled water, and dry a t 1.50"C. Hydrochloric acid, 0.04N, and sodium hydroxide, O.O4N, for adjustment of pH of distillate to between 3.05 and 3.10. Standard Sodium Fluoride Solution. Dry sodium fluoride for 2 t o 3 hours in a n oven a t 110" C. Cool in a desiccator. Dissolve 2.2108 grams in 1 liter of distilled water. Take a 10-ml. aliquot and dilute t o 1 liter. Store in polythene bottles. The latter solution contains 10 y of fluoride per ml. It is best to check the purity of the sodium fluoride by converting samples
into the sulfate; apply appropriate correstions t o weighed samples. Thorium nitrate, 0.001M. Dissolve 0.276 gram of thorium nitrate tetrahydrate in 500 ml. of water. It is advisable to check the titer of this solution once a month. SPADKS, 0.020%. Synthesize the dye as directed by Banerjee (5'). Dry a sample in an oven a t 110" C. for an hour and cool it in a desiccator. Dissolve 0.200 gram in 1 liter of water. The solution is very stable; a stock solution has been in use for over 6 months without change. Use reagent grade cliemicals throughout. ANALYTICAL PROCEDURE
Grind a rock sample by percussion mortar and pestle to pass a 90-mesh sieve. Dry the powder in an oven a t 110" C. for an hour and cool it in a desiccator. Weigh a 0.25-gram sample t o within 1 mg. into a platinum or nickel crucible. Add eight to ten times the sample weight of sodium peroxide and mix thoroughly with the sample. Place the crucible and sample in a muffle furnace a t 270" t o 300" C. Bring the temperature up over 30 minutes to 480" to 500' C., and hold a t this temperature for 20 to 30 minutes. Cool t o room temperature, dissolve the frit in 100 ml. of distilled water, mash the crucible with a little hot water, and add the washings t o the solution. Simmer the solution for an hour to remove excess peroxide. Add don-ly with stirring about 5 ml. of zinc sulfate solution and 3 drops of the indicator solution, and adjust t o p H between 11 and 12 by dropwise addition of 1 to 3 sulfuric acid. At this p H the indicator is nearly colorless. If the pH is too low, readjust by addition of 207, sodium hydroxide. Digest the bulky precipitate of zinc silicate, zinc aluminate, and hydroxides on a steam bath for about an hour. Filter off on IVhatman 54 filter paper and wash the precipitate several times with hot 1% sodium sulfate solution. Collect the filtrate and washings in a 250-ml. beaker and evaporate to about 20 ml. on the steam bath. Place 20 nil. of sulfuric acid with about 0.3 gram of the treated soda-lime in the distillation chamber of the still. Add the evaporated filtrate, and rinse out the beaker into the chamber with hot xater. RIix the sample with #,he sulfuric acid by admitting air bubbles from the steam generator. When hhe sum-tetrachloroethane refluxes, admit steam to the distillation chamber and collect the distillate in a 100-ml. volumetric flask a t a rate of 1.5 to 2 ml. per minute. After the first 10 minutes increase the production of steam so that distillate is collected a t a rate of 2.5 to 3 ml. per minute. Collect about 98 ml. of distillate. Adjust the pH of the distillate t o between 3.05 and 3.10 by means of a pH meter and the O.04N aaid or base. Makp up to volume with washings of $he electrodes and beaker used for the Bdjustment.
Transfer into the Perspex cuvette an aliquot containing not more than 70 y of fluorine (25 ml. is usually a convenient aliquot), and add 0.50 ml. of the 0.020% dye solution. Titrate spectrophotometrically with standard thorium solution. r o t e the change in optical density after each 0.050-ml. increment of titrant. Once past the end point, add the titrant in 0.100-ml. increments. Determine the end point graphically by plotting absorbance against volume of thorium solution. Draw the slopes of the two lines as described by Bricker and Sweetscr (4). Take a second aliquot and titrate as described. Average the two results and calculate the per cent of fluorine present. The two results should agree within 0.018 t o 0.020 ml. of titrant. Standardize the thorium solution as follows: -4dd known amounts of the sodium fluoride solution t o the distillation chamber and carry through the procedure from there. The titer should be about 0.18 t o 0.20 ml. of thorium solution per 10 y of fluorine. Standardization by carrying alkali fluoride through the distillation step is adopted for tiyo reasons: (1) Distillation by the procedure described produces fluosilicic acid, the reaction of which with thorium nitrate is different from that of alkali fluoride ( 1 9 ) , and (2) a constant amount of fluorine remains behind in the distillation chamber. Solutions of reagent grade fluosilicic acid, standardized by the lead chlorofluoride method ( 8 ) , were titrated with the thorium nitrate; the ratio of the titer for solutions of distilled fluorine to the titer for solutions of the standardized fluosilicic acid indicated that recovery is 98% over the range of 0 to 200 y of fluorine for the distillation apparatus in use. Determine the blank on the reagents, and establish the sulfate interference for the conditions of distillation employed. The blank on the reagents is the sum of fluoride impurity and the effect of sulfate entrained in the distillation. Blanks have been from 0.04 to 0.05 ml. of titrant. For a titrant of the same strength different blanks are obtained only when new stocks of sulfuric acid are introduced. Replicate determinations on the same set of reagents have given a maximum deviation of 0.004 ml. of titrant.
EXPERIMENTAL
The experiments were designed t o establish the following: The control of pH in the titration of fluorine as fluosilicic acid. The range of fluorine that can be titrated without serious variations due to salt effects or changes of pH. The recovery of fluorine added in known amounts of synthetic rocks carried through the procedure, and the volume of distillate required for acceptable recovery. Interferences from elements expected in normal rocks.
Comparison of results on standard rocks with results reported by other workers using other methods. Reproducibility of the method. pH is known to affect the titer when fluorine is titrated with thorium in the presence of a hydroxyazo dye. Wadhwani (19) has studied the p H effect on the stoichiometry for the thoriumsodium alizarin sulfonate-fluorine system. His explanations of the effect of p H on that system probably apply t o the thorium-SPADNS-fluorine system as well. Moreover, Wadhn ani has shown that the reaction of fluorine as fluosilicic acid with thorium is ( 1 9 ) :
+ + + +
Th(S03)d H2SiF6 4H20 = HzThFB 4HS03 H4SiOd (1)
Evidence given by TVadhTT-ani ( 1 9 ) suggests that a t the pH range 2.9 to 3.1 no further reaction betneen thoriumnitrate andfluothoric acid (HzThF6) to form thorium tetrafluoride takes place.
Figure 2. Titration of fluoride in chlorite schist with standard thorium nitrate 0.206 ml. = 10.0 y F; wave length, 580 mp
The production of acidity in this reaction will affect the stoichiometry of the system by lowering the pH (20). Consequently, a buffer was sought that would keep the pH within about 0.2 unit, so that the titer would not be considerably affected. However, the hydrochloric acid-sodium acetate buffer used by Banerjee ( 2 ) was unsatisfactory, as were phthalate, citrate, and monochloroacetic acid buff.ers because acceptable p H control was obtained a t the sacrifice of sharp color change a t the end point. Studies on the variation of pH after adjustment of distillates with acid or base showed that for moderate amounts of fluorine the production of acidity (Equation 1) is not great enough to produce a p H decrease of more than 0.2 unit. However, as the solutions were not buffered, the range of fluorine that could be titrated would be limited in proportion to the acidity produced. It was found that the titer remained linear up to 75 y of fluorine; for larger amounts, the titer increased (Table VOL. 2 9 , NO. 8, AUGUST 1957
1131
Table I.
pH Control by Acid-Base Adjustment of Distillates
pH before pH Adjusted pH a t End of Titer, All. Adjustment to Titration Th/lO 7 F 3.56 3.02 2.96 0.190 10.0 3.28 3.02 2.98 0.188 3.06 2.94 0,190 25.0 3.18 75.0 3.42 3.08 2.94 0,188 125. 3.16 3.09 2.90 0.199 11.1 2.79 3.04 3.01 34.1 2.82 3.05 2.98 14.5 3.42 3.08 3.00 29.0 3.19 3.10 3.02 Samples 1 through 5, standard sodium fluoride solutions; 6 through 10, rock samples, AG and AB. 7/25-h11.
Aliquot 5.0
I). The system does not obey the laws of chemical proportion at pH 2.9 to 3.1; theoretically, on the basis of Equation 1, the volume of thorium required would be less than half of what is actually used. Such lack of stoichiometry, however, renders possible semimicrotitration of microgram amounts of fluorine. The recovery of fluorine from rocks was tested by adding known amounts of fluorine to synthetic rocks. A rock of granitic composition (AG) was made up from the oxides of silicon, aluminum, ferric iron, calcium, magnesium, titanium, phosphorus, and manganese and from sodium carbonate and potassium sulfate. The powders were thoroughly mixed and passed twice through a 120-mesh sieve. A rock of basaltic composition (AB) was similarly prepared. Blanks of each were made in duplicate, and found to be rather high. Determinations of the individual constituents showed that fluorine was present in the calcium oxide, magnesium oxide, and aluminum oxide, and all the fluorine in the synthetic rock could be accounted for by calculating the amount contributed by each constituent. The compounds used were not reagent grade. Fluorine was added to 0.250-gram samples as an alkaline Table II.
Fluorine Recovery from Synthetic Rocks and Bureau of Standards Soda Feldspar Sample No. 99
Sample No. AG-1 AG-2 AG-3 AG-4 AG-5 AG-6
AB-1 AB-2
AB-3 AB-4 AB-5 AB-6 SF-1
SF-2 SF-3
SF-4
a
SF-5 SF-6 Reported
1 132
solution. After careful evaporation, the samples were carried through the analytical procedure. Known amounts of fluorine as solid sodium fluoride were added t o samples of Bureau of Standards soda feldspar sample 99 by weighing the sodium fluoride on a microbalance. These were carried through the procedure also (SF). The results are given in Table 11. With one exception, the determinations are well within 1 y per aliquot. ,411 the fluorine was recovered in the first 100 ml. of distillate. Ions forming a stable compound or complex with fluorine, thorium, or the dye will interfere. The effects should be similar to those reported for other thorium-azo dye-fluorine systems. Studies by Revinson and Harley (17)) Lambert ( 1 2 ) , and Shell and Craig (18) suggest that for normal rocks analyzed by this method, ions causing interference are chloride, sulfate, phosphate, silicate, aluminum, and boron. Chloride can be removed by distilling in the presence of silver sulfate (10). The effect of sulfate is kept constant by establishing reproducible conditions of distillation (10). It has been found that 0.3 to 0.5 mg. of sulfate per 25 ml. can be tolerated in the titration without significant change in the blank (Table 111). Consequently, a reference
F Added, y 0 . 0 (blank) 0.0 (blank) 24.4 48.8 97.8 195.6 0 . 0 (blank) 0 . 0 (blank) 14.8
78.4 44.0 195.6
0 . 0 (blank)
F, 7/25 311. 0.0 0.0 6.1 12.2 24.9 48.9 0.0 0.0
3.7 19.6 11.0 48.9
0.0 0.0
F Found, y / 25 M1. 11.8 12.0 5.3" 11.1a 24.4" 48. 4a 22.1 22.7 4.3" 19,3" 10.74 48.3" 17.7
Table 111. Effect of Certain Ions on Determination of Fluorine
(37.6 y F added with 25 ml. of water) Amount I on Added, Rlg. Error, y F
SO,--
PO,---
.. ..
13
0.1
0.1 0.2 0.3 0.3 0.4 0.5 0.5 0,025 0.050
0.075
70Error
c1-
BOs---
0.100 0.7 4.0 2. I
0.0 0.0 0.0
$3.6 +4.1 +3.4 +3.5 +4.1 +1.5 +3.6 +11.3 +18.7 No end point +3.4 -3.7
9
2
1
..
16 2
3 1
..
18.3 .. O.O(blank) 16.6 4.1 4 4.34 8.7" 32.8 8.2 6 18.2a 2 73.9 18.5 0 43.2" 173.4 43.4 as fluorine found in addition to average fluorine present in blank.
ANALYTICAL CHEMISTRY
solution containing 0.6 mg. of sulfate and 1.0 ml. of a saturated solution of barium chloride in 50 ml. of water and another with 1.0 mg. of sulfate are used for comparison with distillates similarly prepared (1.O ml. of the barium chloride solution is added to the 50 ml. of distillate remaining after titration of two aliquots). After the solutions prepared have stood for 15 minutes, the standards are held with the distillates against diffuse light to ascertain nhether the amount of sulfate in the distillate is nithin these limits. In over 60 distillations only t t r o distillates contained sulfate in excess of 0.5 mg. per 25 ml. Howeyer, no correction can be consistently applied to titrations in the presence of excebr sulfate, for graphical interpretation ia difficult, as the results in Table I11 show.. The determination must be repeated. The sulfate present in distillates has never been less than 0.3 mg. per 25 ml. If calcium and phosphate ions are both present, it can be expected that they will form fluorapatite, which will be precipitated with the silica and alumina. I n order to study this possible effect a phosphatic limestone with 9.8% phosphorus pentoxide and 48.1% calcium oxide was analyzed by this method and the results were compared to those by the lead chlorofluoride method described by Groves (8). The data in Table IV suggest that
Table IV. Recovery of Fluorine from Rocks Containing Certain Interfering Elements
Fluorine, yo Rock Expected Found AG-7 with 40 mg. of B,O, added 0.022 0.023 Chlorite schist with 40 mg. of B203 added 0 051 0 052 Phosphaticlimestone 1 22a 1 33 Fluorine determined by lead chlorofluoride method (8).
there is no such interference and that the peroxide decomposition is more efficacious. Forty milligrams of boric anhydride was added t o samples of the synthetic granite and a metamorphic rock; the usual procedure was followed (Table IV). Ahrens (1) has given data on the determination of fluorine by several workers on two standard rocks, G-1 and W-1. The agreement is not very good; however, his recommended value (the average of all determinations) is close to the duplicate results obtained by the SPADNS method (Table V).
Table V. Comparison of Results on G-1 and W-1 by Different Workers and Methods
Rock G-1
Method Spectrographic
Result, P.P.M. 900 800
Chemical
450 900 900 400
800
SPADKS
770 800
W-1 ’
a
Av. (Ahrens)” Av. (SPADNS) Spectrographic SPADNS
740 790 200 200
Data from Ahrens ( 1 ) .
As the recovery data indicated an error of about 2y0for aliquots containing about 30 y of fluorine, acceptable accuracy was expected for the microdetermination of samples with large amounts of fluorine (greater than 2.4% of fluorine). For this study Bureau of Standards sample opal glass, KO, 91, was analyzed in two ways: in Procedure A the sample was, after peroxide decomposition, dissolved in about 50 ml. of water, and silica and alumina were precipitated with 1 to 1 zinc sulfate solution. I n Procedure B the frit was dissolved in about 15 ml. of water and the solution was added to the distillation chamber. The results for both methods are given in Table VI. Replicate results for ten metamorphic rocks are given in Table VII. I n the samples which were not run concurrently the analyses were effected with new thorium solution and new
Table VII.
Precision for Replicate Analyses of Some Scottish Metamorphic Rocks
Sample Quartz-mica schist Phyllite Grit Quartz-feldspathic schist Chlorite schist Quartz-mica schist Garnet-biotite schist Quartz-biotite gneiss Quartz-biotite schist Quartz-biotite schist 0.
o”
0.030 0.043 0.042 0.042 0.051 0.057 0.031 0.028 0.040 0.058
0 028“ 0.041 0.044 0. . 041 .-~ 0 . 051 0 . 057 0.033 0.027 0.036 0.058
Procedure A
B
0 041.
...
0.052” ... ... ... ... ... Av.
Deviation 0.001 0.0007 0.001 0.0005 0.0007 0.000 0.001
0.0005 0.002 0.000
0.0007
RIpd Mpd’ 3.4 2.8 1.7 1.9 2.3 1.9 1.2 1.9 1.4 1.5 ... 1.4 3.1 2.5 1.8 2.9 5.3 2.1 ... 1.4 2.0 2.0
Analysis not concurrent with other determination(s).
reagents. The maximum proportional deviation, defined as the maximum deviation divided by the arithmetic mean, is also presented for each of the replicate results. The actual maximum proportional deviation ( M p d ) is compared with an M p d calculated on the basis of an error of 1 y ( M p d ’ ) . Although iMpd is greater than Mpd’ for four results it is considerably greater in only one out of the ten analyses. The average M p d is equal to the average calculated Mpd’ for the ten analyses, suggesting that the method is capab!e of determining fluorine to nithin 4 y per 0.25-gram sample. Modifications Employed. Malachite green oxalate is used in t h e place of a n alkaline blue-1,3,5trinitrobenzene mixed indicator described by Chu and Schafer ( 5 ) . T h e color change of t h e former mas easier t o see i n t h e presence of t h e bulky, muddy colored precipitates. Filter paper has been employed instead of sintered glass funnels ( 5 ) , because suitable equipment is unavailable here for treatment of several samples a t once. Their use a o u l d make easier thorough It-ashing of t h e bulky precipitates. Otherwise, t h e procedure given b y Chu and Pc hafer ( 5 ) is followed. Evaporation in Borosilicate Glass. McClure (13) and Pone11 and Saylor (15) have reported loir iesults n h e n fluoride solutions are evaporated in porcelain 01 glass. T h e data heiein indicate t h a t t h e use of bolosilicate glass for the evapoiation of niicrogram amounts of fluoiine 1‘ qatisfactoiy. The beakers a l e in5pected for scratches piior t o use and cleaned \\ith chromic acid after use. and t h e same ones are used each time, so long as they remain free of sciatches ~
Table VI.
Fluorine
Before a new beaker is used, it is treated with warm chromic acid for 24 hours. ACKNOWLEDGMENT
The author thanks R. A. Chalmers for his help in all aspects of the work. LITERATURE CITED
hhrens, L. H., “Quantitative Spec.trochemical Analysis of Silicates,” pp. 25-7, Pergamon Press, London, 1954. Banerjee, G., Science and Cultitre ( I n d i a ) 20, 611 (1955). Banerjee, G., Z. anal. Chem. 146, 417 (1955). Bricker, C. E . , Sweetser, P. B., A N A L . CHEX. 24, 410 (1952). Chu, C., Schafer, J. L., Ibid., 27, 1429 (1955). Grimaldi, F. S., Ingram, B., Cuttitta, F., Ibid., 27, 918 (1955). Groves, A. W.,“Silicate Analysis,” 2nd ed., pp. 160-2, George Allen and Unwin Ltd., London, 1951. Ibid., pp. 218-21. Hoffman, J. I., Lundell, G. E. F., Bur. Standards J . Research 3, 581 (1929). Huckaby, R. B., Welch, E. T , Metler, A. V., ANAL. CHEM. 19, 154 (1947). Icken, J. &I., Blank, B. M., Ibid., 25, 1741 (1953). Lambert, J. L., Ibid. 26, 558 (1954). McClure, F. J., IND.ENG. CHEV., ANAL.ED. 1 1 , 171 (1939). Monnier, D., Xaucher, R., Wenger, P., Helv. Chim.Acta 33, 1 (1950). Powell, W.A., Saylor, J. H., ANAL. CHEY.25,960 (1953). Analyst 75,485 (1950). Rafter, T. A4., Revinson, D., Harley, J. H., AKAL, CHEM.25, 794 (1953). Shell, H. R., Craig, R. L., Ibzd., 26, 996 (1954). Wadhwani, T. K , J . Indzan Inat. Sci. 34, 123 (1952). Ibid., pp, 138-9. Willard, H. H., Winter, 0. B., IND. ENG. CHEY.. ANAL. ED. 5, -7 (1933).
Microdetermination of Fluorine in O p a l Glass
% F Found 5 67
5 67
Bureau of Standards Result 5 72 5 72
Range, Bur. of Standards Results 5 65-5 80 5 65-5 80
RECEIVED for review December 18, 1956. Accepted April 4, 1957. Based upon a thesis submitted by the author in partial fulfillment of the requirements for the degree of doctor of philosophy a t the University of Durham. VOL. 29, NO. 8, AUGUST 1957
1 1 33
