182 
Mr. Tovey’s ’Researches in the 
and on the length of the wave. Thisj then, is a general re- 
suit from the theory, and it agrees, as we know, with ex- 
perience. 
We shall now proceed with the investigation, in order to 
compare the theory with experiment a little further. 
By (14.) and (23.) we have 
/c = s 4 - k\/ — 1 , ( 36 .) 
and, by transformation, 
= cos a/~ 1. sin^A^; (37.) 
hence, if we put 
(>Qg ^ A,r--1 = w, 
e^^^s\nk^x—u\ (38.) 
we have, by (8), 
s — ^pu + V — 'Sipu' , 
s' = I>p'u+ a/ — 1. I,p' u\ (39.) 
5/==2l5'^^ + a/— - 1 . ^qu. 
If we compare these equations with (20.) we shall find 
(7 = H>pUf a' = ^p'u^ 
(T^ = 2 ^?/, 0-^ = '2>p'u\ (4-0.) 
(T<^~ '^qu ^ (T^ qu' . 
By (13.) and (20.) we find 
(^2 + 0 -+ a/ — 1* <^/) (^^ + 0^' + a/ — 1* O’/) = (o' 2 + a/ — B 0 - 3 )% 
which, since n is real, gives 
{n^+(T) {n^-\-(r') 
[n^ 4 0-) a- 1 4 {n^ 4 <r') <Tj = 2 0-3 . 
Hence, by eliminating n% and reducing, we find 
(2 0-2 0-3— (o-^-o-) (Tj) (2<r^G-2 + ((T'—(r) q'^) 
= (^/^'y + o- 2 -cr 3 ) (o-'^ 4 o-i)S 
which, as appears by (38.) and (40.), expresses, implicitly, 
the relation between s and k. 
To obtain a precise idea of the movement represented by 
the expressions (35.), suppose the arbitrary coefficients de- 
noted by a to be all zero except one ; then each of the sums 
will be reduced to a single term, so that we shall have 
Yj = a sin (nt + k x -i- d) , 
? = (3 a sin (nt 4 kx -h ^ + 7 ) . 
(41.) 
(42.) 
( 43 .) 
