496 The Rev. J. Challis's Reply to the 



cular to the axis of 2", are propagated unaltered in a direction 



parallel to that axis with the uniform velocity a\/ \ -| — _ ; 



in other words, the waves are non-divergent, and the velocity 

 of propagation is greater than a. 



I shall next maintain that e must have a value different from 

 zero. 



In p. 277 I have come to a concUision which has not been 

 disputed, and which is expressed in these words : " It is thus 

 shown that the condition that uilx + vdy + 110(12 he an exact dif- 

 ferential, must be satisfied in a manner which shall equally 

 apply whatever be the original disturbance of the fluid." The 

 supposition that u, u, and x<o are functions of the distance from 

 a fixed plane, is one of a general nature, by which udx-\-vdy 

 ■^•wdz becomes an exact differential, and which, consequently, 

 is as much entitled to consideration as that which I have 

 adopted. This is, in fact, the case of plane-waves, and coin- 

 cides with the supposition that b^ = 0. If the motion be pa- 

 rallel to the axis of ^r, 'w= -—, and the exact equation appli- 

 cable to plane-waves is 



dt^ \ dz^ ) dz^ dz dz dt ^ ^ ' 



which, as is known, is satisfied by the equation 



'w-f{:z-[(t-\-xa)t). 



The function y being quite arbitrary, we may give it a parti- 

 cular form. Let, therefore, 



U) = ?« sm — (r — (a + Tt))^). 

 A 



This equation shows that at any time t^ we shall have 'a)=0 

 at points on the axis of ^, for which 



+ m at points of the 



0^' 





