182 Mr. Tovey's Researches i?i the 



and on the length of the wave. Thisj then, is a general re- 

 sult 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 



a; = s + k^/^^l, (36.) 



and, by transformation, 



^M^VTl — cos ]cAx+ v'^^l. sin A"A a'; (37.) 



hence, if we put 



e'^^sin ^Aa;= w', (38.) 



we have, by (8), 



s = S^M^ + V—l. 'S.pu', 

 's' = ^p'u+ \/^. -Ep'u', (39.) 



s, = I, q u + //•— 1. Sgw. 

 If we compare these equations with (20.) we shall find 

 <x = ^pn, a' = l^p'u^ 



0-^ = Sp?/, a',= ^p'u\ (40.) 



(Tg = 2 g' M , a-Q =. "Lqii' . 



By (13.) and (20.) we find 



which, since n is real, gives 



(n^+o-) («2 + (r')-o-,o-', = (T^-a^g, ^^^^^ 



(«2 + 0-) 0-/ + (W^ + (7') <T^ = 2(Tci^<Tq. 



Hence, by eliminating n^^ and reducing, we find 



(2 <T^(TQ—{(T^ — (r) 0-^) (2(72(r3+((7'— cr) a'^) 



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 



Yi = a e^^^ s'm hit + kx -^ b), , . 



(43.) 

 ? = ^ae^^ sin {nt + kx -{- h + y). 



