OSCILLATIONS OF A SUSPENSION CHAIN. 385 



.-. = - 2 — M + h sin p \/cgt sin pa. 



<f v 



Also ^ =0; 



, u v a • ^ F u j j • • <-v a 2w + !t 2b+1ts 

 (•.• by hypothesis — - can be expanded in a series ot z . An cos , 



(t S" <£& £a 



2n + 1 . 7r 



and that 2 An is convergent) ; 



2a 



.•. o = M + hp cos pa sin p y/cgt ; 



c 



.•. = sin pa —pa cos pa, .•. tan pa =pa. 

 This condition is easily shewn to be the same as (7), for putting 



cos 



•-'(--OC'T-^t'-^r) 



Taking logarithms and differentiating 



20 20 



tan 6 = 



4 4 



Let 20 = irq; 



vq ttq . 



tan ir = i % 2 + &c - 



2 7T Tr 2 ^ 2 



t\1- 



1 



+ 



g 2 3" - q' 



And i^7 + *.&-?) + ¥{^T) + " ° may be written 



— Is. I + + &c> = 0; 



q*\ (2» + l)" \l-q 2 3'-q* J 



1 1 „ 7T 8 7T ■7rQ' 



li + ~, — ~, + &c. = — = — tan -2- ; 



1 -q* 3> - q 2 8 4q 



■n-q irq 

 .". tan — = — , 

 2 2 



which is identical with pa = tan pa, since pa = — . 



The types of vibration I had found for A 3 were \/8 .2 .eg. — and v 24. 2. eg — , and 



I proved that the greatest amount of "play 11 at the end from the neglected terms A i} A i9 &c. 

 would be little more than -j^th the maximum disturbance at the centre of the chain, of course 

 this small quantity might be safely set aside, and we shall now see that the accurate values of 

 the types, corresponding to those obtained at first, differ imperceptibly from the above values. 



