﻿of Elastic Stresses in an Isotropic Body. 283 



and if we write P', Q', R' for the Particular Integral portion 

 of P, Q, R — i. e. the terms which depend explicitly on g — 



we find 



11X — it 



?'=ffp\x + ~^gp(W + vz), . . . (12) 



with similar values for Q' and R', the Complementary 

 Function part of P, Q, and R and the values of S, T, U 

 being those given in the earlier part of this paper. 



The other case which I propose to consider is that of 

 a body moving with angular velocities co x , co y , co z about 

 the axes of coordinates which must be axes fixed in the 

 body. It is implied either that the question is purely 

 kinematica], or that a problem in Rigid Dynamics has 

 been previously solved. 



The expressions for the acceleration of a point in the 

 body are well known, and give for the effect of the reversed 

 effective forces 



F = 



i{0)/ + »/> 2 + (w 2 + ft>*V+ (o> x 2 + a)/)* 2 



— 2(o x (D y xy — 2co x o)zxz — 2co lJ (t) y yz } , 

 with 



f l = x(yw,—z6)y), fz—y{z<d x —!cwz), f 3 = z(xw y -y6) z ). 



The form of the forces / x , / 2 , / 3 indicates that they will 

 cause no strain in the body, and consequently cause no 

 stress. If we proceed to find the effect which they have on 

 the values of the stresses, they will be seen to disappear from 

 the stress-equations. I shall therefore omit them for this 

 purpose, and treat %/, yj > X$ as nu ^- 



We then find 



*' = ~ 12(Jt n) { (< + W * 2) * 4 + (ft>/ + "^ 4 (W/ + "^ 



- 2co y co z yz(y 2 + z 2 ) - 2co x co z xz(x 2 + z 2 ) 



— 2(D x (0 y xy(x 2 + y 2 )} 



s '=Hi' ete -' w 



with Complementary Function terms as before. 



These give no solution of any specific question. They 

 only give a skeleton of the general form which a solution 

 will take. 



and 



