140 Prof. Challis on ike Mathematical Theory of 



the laws of the free motion which takes place along the axis of 

 rectilinear motion ; and as no case of arbitrary disturbance is 

 under consideration, the steps of the analytical reasoning should 

 conduct to a function of explicit form indicative of a special 

 kind of motion. We have next to inquire whether such a result 

 is dcducible, and by what process. First, it may be remarked 

 that, as the equation seems not to admit of being integrated when 

 all the terms are retained, we must proceed by successive ap- 

 proximations, commencing with the terms of the first order with 

 respect to cf>. Thus we shall have to integrate 



S-.*S+i*-a 



df d. 



(6) 



It does not appear that the general integral of this equation can 

 be expressed in a finite number of terms ; the only form it is 

 known to admit of is the following, in which, for the sake of 



brevity, e is put for — v u for z — at, and v for z + at : 



e q v 2 -n , v e 3 v 3 



+ Gfy) + e^v) + g . B,M + j-^-g . G s (0 + &e. 



where 



and 



G 1 (v)=JG(i/)^, G 2 (v)=JG,(v)^&c. 



As the arbitrary functions F and G satisfy the equation inde- 

 pendently, we are at liberty to consider them separately. Sup- 

 pose G to disappear. Then the value of ^> has this peculiarity, 

 that it is an expansion by Taylor's theorem of a finite expression, 

 if forms of F can be found which satisfy the equality 



for every integral value of n. Now since F w+1 (m) = §¥ n (u)d/jL } it 

 follows from the above equality that 



d 2 F (u) 



As the upper sign would lead to a logarithmic form of F, which 

 is incompatible with any general law of fluid motion, we must 

 take the lower, and shall then have by integration, 



F rt (w)=Acos(V + B). 



