PROFESSOR KELLAND ON THE THEORY OF WAVES. 141 



Hence 



/" ae f" 1 



(p cos w e J rf<p = l + -H- / e 



\ } 



a )\ 



By putting -(a-d*) and a -' successively for r, and subtracting the results, 

 we shall obtain the terms corresponding to a a' in -~^-, and by adding to this 



result a similar expression with cC instead of a a', we shall obtain the whole of 

 the expression under the integral sign in equation (a). 

 Now the first difference is 



3 3 



2 4 w fa /, ws/ 1 ~<T r n> J 1 % \ 

 -T=- . ^5 - I I + s / , e da\ 



vg-t fft \ ' y ~ V i / 



and the second 



2w 4W0 1 / wv i ~2 / "" " "~a 



I i - f i f> 



' 9? 



- ^ 



1 + - _ e da 



) 



dd) 



- 2'TT ' 



r <*' - r a - 



/ ~"2"j e 2 I e * 



o/ c da= --- / - 5 



\r 2j e e da 



WOW o/ c da= --- / - 5 



a a 8 



a' a' a? <f_ . , 



" ~ ~" 5 e ~2 



+ - &c. 



2 T - 1 1 3 1 . 3 . 5 



e * da 



2 T /-, 1 1 3 1 . 3 . 5 \ 



-- e (1 + -^+ j- + - r -f &c. ) 



V-1 V w w w ' 



_ 1 _ 1 1.3 \ 

 w a W 4 -y 



Hence, as the time increases, the velocity tends to vary inversely as the cube of 

 the time ; a result obtained by very different processes as well by M. POISSON as 

 by M. CAUCHY. 



II. Let us next suppose that the motion has been produced by an impulse at 

 the surface of the fluid, but without any elevation or depression of the surface'. 



Hence, at the surface, when *=0, $=/(a), and ^?=0 



VOL. XV. PART I. P p 



