ON STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 207 
as to take out accidental inequalities. A check on the accuracy of the 
calculation is easily provided, for the calculated P—Q should agree with 
the value optically observed. 
In many problems it is known that one of the normal stresses is through- 
out verysmall. In this case, if Q, say, is nearly zero, we have P=R cos 2a, 
and the stress difference leads easily to the complete system of stresses. 
This assumption has been made by Coker in his earlier papers, but it 
would seem desirable to justify it more fully. 
NOTKS. 
(References to these are given in the text.) 
(1) Karl Pearson, A. F. C. Pollard, C. W. Wheen, and L. F. Richardson : 
An Experimental Study of the Stresses in Masonry Dams. (Drapers’ 
Company Research Memoirs: Technical Series V.) 
(2) J. S. Wilson and W. Gore: Stresses in Dams. ‘ Proc. Inst. C.E.,’ 
1908. 
(3) H. N.daC. Andrade : The Distribution of Slide in a Right Six-face 
Subject to Pure Shear. ‘R.S. Proc. A.,’ vol. 85, pp. 448-461. 
(4) Sir David Brewster: ‘ Phil. Trans.’ 1816, p. 156. ‘ Annales de 
Chimie et de Physique,’ vol. xx. Fresnel : ‘ @uvres d’Augustin Fresnel,’ 
tome 1, p. 713. F. E. Neumann, ‘ Abh. d. k. Acad. d. Wiss. zu Berlin,’ 
1841, vol. ii., p. 50-61. See also ‘ Pogg. Ann.’ vol. liv. John Kerr: 
* Phil. Mag.,’ 1888, ser. 5, vol. 26, No. 161. G. Wertheim: ‘ Annales de 
Chimie et de Physique,’ ser. 3, vol. xl., p. 156. 
(5) F. Pockels: ‘Ueber die Aenderung des optischen Verhaltens 
Verschiedener Glaser durch elastische Deformation,’ Ann. d. Physik, 1902, 
ser. 4, vol. 7, p. 745. L.N.G. Filon: On the Variation with the Wave- 
length of the Double Refraction in Strained Glass, ‘Camb. Phil. Soc. 
Proc.,’ vol. xi. Pt. vi., vol. xii. Pt.i., and vol. xii. Pt. v. On the Dispersion 
in Artificial Double Refraction, ‘ Phil. Trans. A.,’ vol. 207, pp. 263-306 
(1907). Preliminary Note on a New Method of Measuring directly the 
Double Refraction in Strained Glass, ‘ R.S. Proc. A.,’ vol. 79, pp. 440-442 
(1907). Measurements of the Absolute Indices of Refraction in Strained 
Glass, ‘ R.S. Proc. A.,’ vol. 83, pp. 572-578 (1910). On the Temperature 
Variation of the Photo-elastic Effect in Strained Glass, ‘ R.S. Proc. A.,’ 
vol. 89, pp. 587-593 (1914). 
(6) Clerk Maxwell: ‘ Trans. Roy. Soc. Hdin.,’ vol. xx., 1853, p. 1172 ; 
or ‘ Collected Papers,’ vol. i. 
(7) A proof of the statement in the text is as follows :—Let E be the 
stress function for generalised plane stress (Love: ‘ Theory of Elasticity,’ 
pp- 86 and 446), P, Q, S the mean stresses 22, 77, 7 in the usual notation, 
R the principal mean stress-difference, ¢ the angle which the lines of 
principal stress make with the axes. 
Then it is known that 
R= (P—Q)"-+48" 
tan 24=28/P—Q 
25=P sin 2¢ P—Q=R cos 2¢. 
Also the mean stresses are given in terms of the stress function by 
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