﻿376 Mr. C. Chree on the Relation between the 



where we employ the symmetrical notation of Todhunter and 

 Pearson's ' History,' and refer everything to fixed Cartesian 

 axes. Also if V. yi ', V denote the direction-cosines of the 

 outward-drawn normal at any point on a surface, and F, (x, H 

 be the components of the applied force there per unit area of 

 surface, we must have 



\' xx + fJ xy + v'xz = F, J 



*"* *■"* '~ s L (7) 



\ ! xy +/jJyy + v'yz = G, [ • w 



X'xz +fi'yz +v ! zz = ~H.J 

 The internal equations are obviously satisfied by 



xy=yz = xz=0, -^ 

 xx =yy = zz = — p, J 



(8) 



where p is any constant. 



If the material be bounded by one surface S e , this is ob- 

 viously the solution for a uniform pressure p over that surface, 

 for the components of p are 



F = -X'f>o 



&=-t*%\ (9) 



H=-i/p,J 



so that the equations (7) are satisfied. 



Answering to (8) we have everywhere a uniform " dilata- 

 tion " A given by 



A=-p/k, (10) 



where k is the bulk-modulus *. 



As A is uniform, the reduction 8Y e of the volume V e 

 enclosed by S e is given by 



8Y e /Y e =-A=p/k, 

 and so 



pY e k ^ iL) 



Suppose, now, auy imaginary surface S* drawn in the 

 material enclosing a volume V*. Then since (8) holds every- 

 where, it holds over the surface S,-, and so glancing at (7) we 

 recognize that the stress over S* is a uniform normal pres- 



* See Thomson and Tait's ' Natural Philosophy,' Part ii. art. 682, 

 and Love's * Elasticity,' vol. i. art. 41. 



