132 Mr. Challis on the General Equations 



whatever as » is an arbitrary function) the velocity at the vena 

 contracta is that acquired by falling through its depth below 

 the surface of the fluid, whatever be the shape of the vessel, 

 or the nature and position of the orifice. The exact con- 

 formity of this result to experience is a proof of the justness 

 of the reasoning from which it is deduced. 



In general ^ .= f{t) + -^ ; -J" = _ _ ^ = j ; /^ = 



^^ = ,. Hence ^^=-I^^-2,^ andlf=/'(0- 



^^ - 2 f. Therefore ^ = V + !^^ + 1} -/' (0, an 



equation which embraces every kind of motion, and in which 

 r is a function of x, y, z, and t, always given by the given 

 conditions of the problem to be solved. 



It would be easy to multiply examples: the preceding 

 suffice to show that the integral we set out with is the proper 

 general integral of the differential equations of the motion of 

 incompressible fluids, and to give an idea of the mode of em- 

 ploying it. This method of deducing the laws of physical 

 action from the integrals of partial differential equations, is 

 new, I believe, and at the same time, important, as it extends 

 to the more interesting problem of the small vibrations of 

 elastic mediums. By parity of reasoning Euler's integral of 



the equation j^ = ^' {^, + df^ + I^) '^ ^he general mte- 

 gral. This point I have considered in a paper lately sub- 

 mitted to the Cambridge Philosophical Society, and have 

 employed this integral in the solution of some problems 

 hitherto not subjected to analysis. The foregoing discussion 

 also throws considerable light upon the nature of the arbi- 

 trary functions which occur in the integrals of partial diffe- 

 rential equations. It may be inferred from it, that while we 

 justly admit the discontinuity of these functions, that is, their 

 changing abruptly from one form to another, it is neither 

 necessary nor proper to suppose the existence of a new species 

 of functions, discontinuous per se, and by this property of dis- 

 continuity distinguished from every other. When Lagrange 

 had shown that the vibratory motions of the particles of elastic 

 mediums were not subject to any law of continuity, it was 

 perhaps too hastily concluded that they must be given by a 

 new order of functions ; for it is not logical to draw infei'ences 

 about physical laws from functions, the existence of which 

 cannot be proved by pure analytical reasoning. This ques- 

 tion, 



