280 Prof. Challis's Theoretical Determination of 

 The velocity in the direction of 2; \sf-j-. Hence 



=f*/siny (^-«^\/l + ^ +c') ; 



tt): 



also 



It hence appears that the velocity of propagation of the wave 

 whose breadth is X, is 



The value of e depends on equation (4.). 



Since equation (1.) is linear with constant coefficients, it 

 will be satisfied by the sum of any number of such solutions 

 as that just obtained, _/, e, [/., K, and c being different in general 

 for each. Hence we have generally, 



where W = S(w) and S=S(s). 



It follows, since in each of the terms under the sign S the 

 quantities which are independent of 2f and t are at our disposal, 

 that we may satisfy by this integral any state of the fluid in 

 the direction of z, subject to the limitation that the condensa- 

 tion and velocity are at all times small. The course of the 

 reasoning shows that the particular form of the function G, 

 which has conducted to the above results, has not been arbi- 

 trarily adopted, but is really the only form that determines 

 the velocity of propagation, and gives a definite solution of 

 the problem. Also as the particular supposition by which 

 udx + vdjz + ivdz was made an exact difi^erential, has conducted 

 to the above values of W and S, which are of general applica- 

 tion, no want of generality has been introduced by that sup- 

 position, so far at least as the motion in the direction of ^ is 

 concerned. I proceed now to the consideration of equation 

 (4.), by which the motion transverse to the axis of s is defined. 



As this equation does not contain t, there is no propagation 

 of motion in any direction parallel to the plane of ,23/; or the 

 propagation in the direction of z takes place without lateral 

 spreading. A value ofy expressed in finite terms is not there- 



