320 Mr. Challis on Small Vibratory Motions, §'C. 



connected together by no law of continuity. If therefore the 

 quantity sought after in any physical question, be given by the 

 solution of a partial differential equation, this circumstance is 

 itself a sufficient proof that the quantity is not subject to the 

 law of continuity : it is a proof too that several quantities of 

 the kind sought for may coexist. But of the infinite number of 

 functions that will satisfy the differential equation, there will in 

 general be a certain species which belongs in a peculiar manner 

 to the question, and is to be determined by a discussion similar to 

 that with which we commenced our consideration of the vibra- 

 tions of an elastic fluid. No general rule can be given for such 

 a discussion ; the nature of the question itself must decide the 

 manner of conducting it. It is essential that this primary form 

 of the arbitrary functions be ascertained, before any application 

 be made of the integral. 



The views contained in this paper, T have found to be 

 greatly confirmed by similar reasoning applied to the general 

 equations of the motion of incompressible fluids. 



J. CHALLIS. 



Trinity Colleoe, 

 March 30, 1829- 



