340 Dr. 0. Cbree on Longitudinal 



we obtain the two results 



E(A 1 +3A 3 *^+A 3 'V + ...) 



= - W 9 1 3 (C + C 2 * 2 2 + CV V + ...)+ 9B1V - -W + . . . | , (8) 



E(B 1 ' + B 3 V + 3B 3 'V+...) 

 _ -k 9 P ^ (C + (W + C 2 % 2 + ...) + V Ai/c 2 2 - B x V + . . . I . (9) 



The terms not shown are of the order /q 4 , k 2 4 , or higher 

 powers of tc x and # 2 . Combining (8) and (9) we get 



E(A 1 -B 1 0=^p(l+7 ? )(A 1 ^ 2 -B 1 V)+..., . . (10) 



E (A, + B/) = - 2k* P ( v >p) (C + C 2 * 2 2 + Co V + . . .) 



+ (1-9,)^(A 1 / C2 2 +B 1 V) + ... • (11) 



To see the significance of (10) replace k^p by its approximate 

 value /> 2 E , when we have 



A 1 -B/=(1 + ,){A 1 0« 2 )*-B 1 >* 1 )*}+.. • (12) 



As we have seen, p=(i + J)w/Z; and thus, so long as i is not 

 too large, {pfc 2 )- and (/^J 2 are in a thin rod small quantities 

 of the orders {kJI) 2 and (/cjl) 2 . Hence we deduce from (12) 

 as a first approximation 



B/ = A ; (13) 



This is all we require for our present purpose ; but, in pass- 

 ing, it may be noted that as a second approximation we have 



B^Mi+a+^Vi 2 -*/)}. 



The more the section departs from circularity — i. e. the 

 more elongated it is in one direction — the greater is the dif- 

 ference between B/ and A x . This and the fact that ik x jI and 

 IkJI must both remain small are useful indications of the 

 limitations implied in the method of solution. 



Employing (13) in (7) and (11) we have, neglecting 

 smaller terms, 



(Bp-Afy&O (Co + CW + CoV) = -PprjA^ + K*), 

 2k*p Wp) (C + C 2 * 2 2 + C 2 V) = - 2E Aj. 



Whence we deduce at once, without knowing anything of 

 the constants C 2 , A 3 , &c, 



Wp-k* P /p)+(2P(»,/p) =t?M<h* + *i*)/(2E). 



