RECIPROCAL FIGURES, FRAMES, AND DIAGRAMS OF FORCES. 33 



If we put 



d 2 A d 2 B d 2 C , d 2 d 2 cl 2 A2 



~M + If + 37 =p ' and fa 2 + 7hf dJ 2 ~ A ' 



this equation becomes 



(k + ^n\fA 2 A + A 2 B + A 2 C S \-fk + 1 ^oA2p-2nY=2nA 2 A > . (7). 



We have also two other equations differing from this only in having B and 



C instead of A on the right hand side. Hence equating the three expressions 



on the right hand side we find 



A 2 A = A 2 B = A 2 C = D 2 , say, .... (8), 



(3k + n)2p = (3k + 2?i)3T> 2 -2nV, .... (9), 

 and 



_ - ^ 9&D 2 -2V 07 p-SV 



These equations are useful when we wish to determine the stress rather than 

 the strain in a body. For instance, if the co-efficients of elasticity, k and n, are 

 increased in the same ratio to any extent, the displacements of the body are 

 proportionally diminished, but the stresses remain the same, and, though their 

 distribution depends essentially on the elasticity of the various parts of the 

 body, the values of the internal forces do not contain the co-efficients of elasti- 

 city as factors. 



There are two cases in which the functions may be treated as functions of 

 two variables. 



The first is when there is no stress, or a constant pressure in the direction 

 of z, as in the case of a stratum originally of uniform thickness, in the direction 

 of z, the thickness being small compared with the other dimensions of the body, 

 and with the rate of variation of strain. 



The second is when there is no strain, or a uniform longitudinal strain in the 

 direction of z, as in the case of a prismatic body whose length in the direction 

 of z is very great, the forces on the sides being functions of x and y only. 



In both of these cases S = and T = 0, so that we may write 



p-^-v u---^- o-^'-v rm 



df U ~ dxdy y ~ dx 2 V • (11} - 



This method of expressing the stresses in two dimensions was first given by 

 the Astronomer Royal, in the " Philosophical Transactions" for 1863. We shall 

 write F instead of C, and call it Airy's Function of Stress in Two Dimensions. 



Let us assume two functions, G and H, such that 



F= _ d 2 G and y _ dm 



dxdy ' ' dxdy ' ^ " J '' 



VOL. XXVI. PART I. I 



