﻿Rotation of Slightly Elastic Bodies. 41 



of approximation by remarking the relation between the co- 

 efficients in rj s jr and rj s +i/r and putting instead of 







as in 7j s l r, 



„ I ,— V / (» + l)X + (2n + ] y 2 „ + o ,,, + A 



Rotation of an Infinite Circular Cylinder of 

 Non-uniform Density. 



We may next deal with the case of a cylinder in which 

 the density is a function of the distance from the axis. 

 Treating the case of the solid cylinder, we may put for the 

 density of the cylinder when at rest, 



N 



*=J(r)=$a n r», 

 o 



and N may haye any value from zero (when the density is 

 uniform) to infinity — in which case the series '%a n r n must be 

 convergent. 



The equation to be solved for a first approximation to the 

 value of 7] is as before, 



d 2 7]/dr 2 + 1/r drjjdr — njr 2 = — aw'-rj (\ + 2/jl) . 



The solution is evidently, 



V =Ar- - ^ 2a M r»+7(( W + 3) 2 ~l). 



Tlie fact that there must be zero radial traction on the 

 boundary surface r = a, yields the condition, 



T 1 =.(\+2/a) d V /dr + \r)/r = Q 

 on r = a, giving 



2(X+/*)A 



S [(n+3)(X + 2f*)+X] a n a*+ 3 /(n+4)(n+2j 



X+2/*o 

 =*r^ S [(fi+4)X+2(« + 3)/*] a„a^ 2 /(w + 4)(n f 2). 



K-t-lfi 



