268 Prof. Challis on the Mathematical Theory 



This equation # being represented, for the sake of brevity, as <I>=0, 

 according to the principle above stated the value of </> which ap- 

 plies to the given circumstances of the problem is required to 

 satisfy the equation 



Since under the given conditions Sr, S0, S\ may be considered 

 to be independent variations, this equation is equivalent to the 

 three 



^ =0 ^ =0 ^-0 

 dr V ' dd U ' dX ~ U ' 



It will now be supposed, as before, that 



<£ = F (r ) cos 2 X sin 2 ((9-1*0, 

 this being not a hypothetical form of the function, but derivable 

 from the given conditions of the problem, as the following argu- 

 ment will show. Resuming the equation (y), differentiating it 

 completely with respect to t, and omitting terms of a higher 

 order than the first, the result is 



Supposing that<£ = 0'+jVOO^ we shall have ^!| = €£ +,fr f {t), 



at tic 



and -r = -? = -r- i ana * f° r tne case i n which this equation ap- 

 dt dr dr u r 



plies to the upper surface of the fluid, so that ( ~ )=0, we may 

 put a for r in the terms of the first order with respect to m. Let 

 ~|^and J*g° represent what ~- and — ^- become by this sub- 

 stitution. Then we have 



= 



h ~dr~ + 2R 3 C ° S Xwn <*(*-W 5«- 



Now this equation must be satisfied independently of particular 

 values of X, 0, and / ; which condition is fulfilled by the assumed 

 value of <£ if ty(t) = 0, or yjr(t) be a constant, which, as we shall 

 presently see, is the case. For then cos 2 X sin 2 (O—fjut) is a 

 factor of all the terms, so that the equation becomes 



0=-GF(«) + ^+VF( a ), 



and can be satisfied by an appropriate value of F(«). From this 

 reasoning it follows that the assumed value of cf> accords with 



