﻿d 

 dr 



or 



Rotation of Slightly Elastic Bodies. 35 



By Young's law we can express T 1? T 2 , and T 3 in terms of 

 the extensions of the cylinder, in the well-known equations, 



T^Xidp/dr-l) + (X + 2fjL)(p/r - 1), 



T, = (X + 2fiXdp/dr-l) + \(p/r-l) 9 



T 3 = \(dp/dr + p/r-2). 



Putting these values of T x and T 2 in the above equation we 

 obtain the result 



,(X + 2,)g-l) + ^-l)] 



-SKS-- 1 )-^^- 1 )-^ 



Qa/r) rf^/Wr 8 + W) [( L V dp/dr)/(\ + 2/i) + {\p/r)/(X + !>)] 



[dp/dr-p/r] = -p<7G> 2 /(A, + 2/x). . (1) 



The value of //, or X, varies from about 8 x 10 8 grammes' 

 weight per square centimetre for steel to about 4 X 10 8 

 grammes' weight for copper. The corresponding densities 

 are about 7 and 9 respectively. Terms involving 



0-/0+2/*) 



are therefore of a smaller order than those which involve 

 coefficients of the form _ ,, _ \ 



2/*/(\ + 2,a). 



Let us write co 2 a — q{\~\-2pb). Then putting 



we can obtain a value for p to any order of approximation 

 which is required. Neglecting, first, all terms involving q 2 . 

 we have the equations for 77, 



p = r + qrj l ; dpjdr =l + q drjjdr ; d 2 p/dr 2 = q dPrjJdr* ; 

 q [l + qvi/r] d?n 1 /dr*.+ llr[l + q(2/idf fl ldr)l(\ + 2p) 



+ i*>ViM/(\ + 2/*) + q (d Vl /dr- Vl /r)] = -qr(l + ? % /r) ; 

 q dfyjdi* (1 + y i/x/r) + 1/r [1 + ? (2/t* d Vl [dr)j(\ + 2/,) 



+ frni/r)/[\ + 2/a)] 5 (d Vl /dr- Vl /r) +qr [1 + ^ r] = ; 

 and since we are neglecting terms in y 2 , the equation for 77! is 



/•-' d 2 rji/dr 2 + r drj 1 /dr—rj 1 = — r 3 , 



D2 



