PROFESSOR KELLAND ON THE THEORY OF WAVES. 131 



and fp =9 (z + *) V? nearly. 



d(b bJq r a cosu. (a of} cos/zo' . 



Also u=-/=-^ I /---= -smctd/J. 



dx IT Jo Jj, VJ.b + h * 



_ _ -v (In, 



. f) sin fji(rf + Vg(z + A) t) + sin p. (af Vff (z -t- ti) t) | ->- 



Now the integral / f^rf/u is ^-whatever be q, provided it be positive. 



*J o [A 



Hence, of the four integrals which make up the value of u, it is necessary, 

 in order that u may be equal zero, that all the quantities under the circular sign 

 should not have the same sign. 



When t = o and xf ^ a 



b I g ( IT 7T 7T TT 



~ 



x' a 

 If t^j, -- ., we obtain as the value 



If 



b ff ( -7T 7T 7T 



~ 



b Iff 



-2 ^e>+A- 



If t^ -- - * OT C r 4 .' r _2[ '"I- 



7T > 7m \l i . T \ n >^ cy ~n 7i "' 



We must bear in mind that our results are mere approximations, and cannot 

 be expected to give any thing more than the nature of the disturbance! 



It appears, therefore, that the different particles of the fluid are not put into 

 motion (at least horizontally) until such time, from the moment at which the 

 raised fluid is set free, as would be occupied by a body moving with a velocity 

 =*/0(z + K) in travelling from the nearest portion of the elevated fluid to the point 

 in question. This is a very important conclusion, and differs altogether from the 

 result obtained by Mr CAUCHY. It must be observed, that the demonstration 

 supposes the fluid free in both directions, a supposition absolutely requisite to the 

 form of the function which has been adopted. 



Again, it is evident that not only is the velocity of transmission of the first 



