﻿Rotation of Slightly Elastic Bodies. 39 



To obtain a second approximation we may put 



f/'-i + ^i/^fW''; 



and therefore 



dp/dr = 1 -f qdr}i/dr + qhirf^dr 



and neglect terms involving ^ 3 . Using, again, equation (1) 

 we obtain, if terms in q* are again neglected, 



d 2 tj 2 /dr 2 + drji/rdr — 7J 2 /r 2 = — rj l9 



or r 2 d 2 rj 2 /dr 2 4- rdrj 2 /dr — r) 2 = — fl^r 8 + r°/8. 



The solution is of the form, 



r j2 = a. 2 r—a l r z /8 + r> 1 192. 



The radial tension T x is given by 



q [(X + 2p) d Vl /dr + X^/r] 4- q 2 [(X + 2/*) <fcfe/<*r + X V2 /r'] . 



The condition of zero radial traction on the boundary 

 therefore yields 



(X + 2p) drj^jdr + Kvi/r + q[(X + 2p) d<n 2 Jdr + X7j 2 /r] = 



at r— a, or since the part independent of q already vanishes 

 at r=a, 



(X + 2/jl) di) 2 ldr + Xrjo/r — 

 at r = a. Hence 



T 1 = 7 (2\ + 3^)(a 2 -r0/4 



+ q 2 [(2\ 4- Sf^aja 2 - ?' 2 )/8 - (3X + 5p) (a 4 - r 4 )/S] , 

 and since 



a 1 = (2\ + 3/A)a 2 /(X + /*) 

 T 1 =q(2X+3fi)(a 2 ^r 2 )^ 



+ V 2 [(2X + 3/*)V(a 2 - r 2 )/8(X + //,)- (3X 4- 5/*)(a 4 -r 4 )/8],, 

 and, finally. 



??2 /r = « 1 /[(2^-H3 A 6)a 2 /(X-r^)-^7 8 



- [(3X 4- 5/*K/0 + /*) - »'*] / 24 x 8 



_ 2X4-3/ , /2X 4- 3/, \ _ 1 / 3X + 5|* , _ A 



~ 8(xTm) V~X+>~ / 8.24 V X+/i j* 



