330 Prof. Challis on a Mathematical Theory 



we shall have 



d^. r<i> 1 



A+yt^.rc^^O. 



^sin^+^(^siu^+^cos^-2^siii^) + AVsm6>=0.(7) 



dr'^ ' 7-2 V dd^ 



Here it may be remarked that if the fluid be incompressible, 

 A: = ; and if it be highly elastic, like the medium through which 

 light is transmitted, this quantity will be very small. Also, 

 since, upon the axis of symmetry of the motion for which ^=0, 

 or ^=7r, the condensation must be a maximum or minimum, it 



follows that -^=0, and consequently -^ = 0, for all points of 



d rch 



this axis. Let, therefore, ^^^=-»|r sin^, -^ being generally a 

 function of r and 6. Then 



^ + ^cot^=Jsm^ + 2,/.cos^. 



Hence, differentiating the foregoing equation with respect to 6, 

 and eliminating (b, it will be found that 



dr^ 



Now, according to a principle which I have already frequently 

 enunciated, to obtain from such an equation as this definite 

 laws of motion depending on the mutual action of the parts of 

 the fluid, it is required to obtain from a genercd supposition, 

 made without reference to a particular instance of motion, a defi- 

 nite form of the function ■^. For instance, suppose that "^^ con- 

 tains r only. Then we have the equation 



which by integration gives a definite form of a/t in accordance 

 with that supposition. This equation, which cannot be exactly 

 integrated, is satisfied, on account of the small value of k, to the 

 degree of approximation with which the reasoning has hitherto 

 been conducted, by the solution 



110= -sin ikr-\-d). 

 r 



Let us now proceed to inquire whether this solution applies to 

 the problem before us. 

 In the first place, we have 



' ^ = - sin ih' + d) sin Q. 

 da r ^ 



Also V=f{t)^=^ .X.msin (nt + e), because the function / is 



