By Professor Pell. 13 



cos (2 r — 1) s y and added together, all the terms on the 



right hand side disappear, except those involving a^, and we hare 



2 ^ {r) cos (2 r — 1) sy = ^ — ^- 

 ^ r \ y ^ y I 2 cos s Y 



r = 1 _ ' 



and adding together equations (5) as they stand 



1 O s=n— 1 ^ 



.-. a:^ = - 2 4> (r) + - E S'^^ (<f> (r)cos (2r — 1) sy) 



s =1 



COS (2 /• — 1) «y coSjM,sif (6) 



ial conditions be, a?r 

 equation (4) 



If the initial conditions be, a?r = 0, ~ = ^\{f)y then from 



cos (2 r* — • 1) sy 



cos sy 



4>i 00 = 5 + 2 is i«-s 

 These equations give as before 



r = n I 



2 (±1 (r) cos (2 r — 1) sy = - — " 



r = i 2cossy 



2 ^1 (r) =^ iih 

 .-. ajr = - 2 <pi 0') t + - 2 ~ 2) ((pi (r) cos (2 r^l) ^y) 



s = 1 



COS (2r — 1) sy . , ,w. 

 ^^ ^— ^ sm|t;.3^ (7) 



which completes the solution to a first approximation. 



It may be observed that the formulae given by Poisson for the 

 longitudinal vibrations of an elastic rod may be easily deduced 

 from the above results. It is remarkable also, that by putting 

 t = 0, anda?j. = f (r) in equation (6), a general analytical theorem 

 may be deduced, of which Lagrange's theorem, that when 

 (2 (0) = and f (a) = 



^^^' = ^2 (J 9 (x) sm—-^ sm —^ 



o 



is a particular case. 



In order to determine how the system would vibrate if dis- 

 turbed, and then left to itself; suppose the first atom to receive 

 a blow impressing upon it a velocity on a, which if the next atom 

 were fixed, would cause it to vibrate through a space a nearly, 

 a being small compared with h. We have then (p (r) = 0, 

 (^^ (r) = ma when r = 1, and zero for all other values. 

 2 cpi (f) cos (2 r — 1) s y = m « sin s y 



2 (pi (r) = m a 

 mat a s=ii-i ^ 



a;. = + - 2 sm (2 r — 1) s y sm fts ^ 



