2 Gwyther, Specificatio7i of Stress. 



2 {yn-n) t Sy 8x) V ; 

 where V 2 = O, &c, v 2 0' = O, &c (3). 



From these we deduce 

 1 

 3 m 



P+Q + R+ -JO— I x JL + .y± + z A\ (p+Q + R) 

 xm-n \ 8x Sy 8z J 



= + * + *- .... (4). 

 I shall suppose (that all the functions are arranged in homo- 

 geneous groups, and shall proceed with the homogeneous groups 

 of order r. 



and 



We shall then have 



((r+3)m-n)(;P+Q + R) r =( 3 m-n)(e + &+*ir) r . (5), 



Pr = Qr~ r—^ x±(e + <f> + *) r , 



(r+$)m-n 8x 



{r+3)m-n Sz 



K = ^r- 1 T zMe + Z + ^r, 



(r+s)m.-n Sz 



Sr = Q'r --T, ^ v(j'/-+»#-)(e + *-+*)r, 



2((r+T ) m)-n)\ 8z 8y ' 



2((r+ 5 )m-n)\ 8x & ' 



u r =^' r - — * J x JL+y±.)(e+*+v) r . (6). 



2{(r+$)m-n)\ 8v 8x ' 



The components of the force per unit volume in the direc- 

 tions of the axes are found on simplification to become 



§e r:+ ^ 1 + 8_^_ (r + 3)m J (e + ^ + y) 

 8x Sy 8z 2{(r+s)m-n) 8x 



«b: + ^ + gev_ y + 3>> A(e + *+*)„ 



Sy &# gs 2( (r+3)m- n) 8y 



Sz Sy 8x 2{ (r+3)m-n) Sz 

 and equated to the proper expression for the particular term in 

 the expression for the fojrce thiesie give the relations between the 

 arbitrary functions. We may take 



SySz SzSx 8x8y 



where X, jx, v, are arbitrary spherical harmonic functions, and then 

 express 0, <J>, ^p in terms of X, fx and v on the lines of Airy's solution, 



