200 RECIPROCAL FIGURES, FRAMES, 



and it, are increased in the same ratio to any extent, the displacements i 

 tlu- 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- 

 thVients of elasticity 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 2, as in the case of a stratum originally of uniform thickness, in the di na- 

 tion 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=0 and T=0, so that we may write 



' dxdy' 



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 



d*G T , 



F= *** V = 



dxdy> dxdy 



then by Thomson and Tait, 694, if a is the displacement in the direction of x 



(13). 

 Case L If # = this becomes 



Integrating with respect to a; we find the following equation for a 



7 ............ (14), 



