ON STRESS DISTRIBUTIONS IN ENGINEERING MATERIALS. 209 
Differentiating (6) with regard to &» and then putting €=0 and 
n=0 respectively, we find 
Bs"(={ 52. | By"(8)-+ nly’) Jeno 
B= { 5 e-+4[ By"(n)+2,'Cn) | bo 
4.€., 
Bn —=4e (58), Ba") tom {8,04 9850) | (9) 
By"()= —4e( 58) Hy") | Hy(0)-+-€8/""(0) f (10) 
Assume E,'’(0)=A, E3’’(0)=B, E,’”(0)=C, E3’’(0)=D. 
Equations (7)-(10) determine E,”(7), Ey’’(n) and hence H,’(£), E,'’(é) 
as homogeneous linear functions of A, B, C, D. 
Hence E=Ae,+ Be,+Ce,+De,+ a€+ (3+ yén+s, where €1, C2, @3, C4 
are now known functions and a, £, y, 6 are arbitrary constants. 
The termsin a £ 6 do not affect the stresses and may be dropped. 
The term y £ 7 may add y to P+Q. 
If, now, the value of any stress be known at a given point, this leads 
to a linear equation between A, B, C, D, y. 
Hence the complete specification of the stress at two points leads to 
six equations for A, B, C, D, yin like manner, if we consider the conditions 
at the boundary, where two of the stresses are in general known, the con- 
ditions at three points give six equations. In either case we have more 
than enough equations to determine A, B, C, D, y. 
Thus the stress conditions at a few points, together with the isoclinic 
lines, determine the stress system completely. 
(8) O. M. Corbino and Trabacchi: ‘ Rendiconti Acad. dei Lincei,’ 
vol. 18,1909. See also letter by O. M. Corbino in ‘ Nature,’ Jan. 16, 1913. 
(9) Volterra: ‘ Annales de l’Ecole Normale de Paris,’ 1907. 
(10) L. N. G. Filon: The Investigation of Stresses in a Rectangular 
Bar by Means of Polarised Light, ‘ Phil. Mag.,’ Jan. 1912. 
(11) Volterra, loc. cit. Note (8); Corbino, loc. cit. Note (7). Filon, 
loc. cit. Note (9) ; also Filon, ‘ Phil. Trans. A.,’ vol. 201, pp. 63-155. 
(12) Carus Wilson: ‘ Phil. Mag.,’ ser. 5, Dec. 1891. 
(13) Boussinesq: ‘Comptes Rendus,’ vol. 114, pp. 1510-1516. See 
also Flamant : ‘Comptes Rendus,’ vol. 114, pp. 1465-1468. 
(14) L.N.G. Filon: Onan Approximate Solution for the Bending of a 
Beam of Rectangular Cross-section under any System of Load: ‘ Phil. 
Trans. A.,’ vol. 201, pp. 63-155. 
(15) O. Hénigsberg and G. Dimmer: Interferenzfarben beanspruchter 
durchsichtiger Kérper. O. Hénigsberg: Unmittelbare Abbildung der 
neutralen Schichte bei Biegung durchsichtiger Kérper in zirkularpolar- 
isierten Licht, ‘International Association for Testing Materials,’ Brussels 
Congress, 1906. 
(16) E. G. Coker: The Determination by Photo-elastic Methods, 
of the Distribution of Stress in Plates of Variable Section, with some 
Applications to Ships’ Plating, ‘Transactions of the Institution of Naval 
"eet See especially pp. 9-11. 
14. P 
