AND THE EQUILIBRIUM AND MOTION OF ELASTIC SOLIDS. 305 



but this is precisely the condition which would have to be satisfied if the fluid had a free surface 

 coinciding with the plane <vx; hence we may suppose the upper half of the fluid removed, without 

 affecting the motion of the rest, and thus we pass to the case of a canal. Hence it is the same 

 thing to determine the motion in a canal, as to determine that in the pipe formed by completing the 

 canal symmetrically with respect to the surface of the fluid. 



We have then, to determine the motion, the equation 



(Pw drw so sin a 

 dai'' ay' fi 



In. the case of a rectangular pipe, it would not be difficult to express the value of w at any point 

 in terms of its values at the several points of the perimeter of a section of the pipe. In the case 

 of a cylindrical pipe the solution is extremely easy : for if we take the axis of the pipe for that of 



z, and take polar co-ordinates r, in a plane parallel to vy, and observe that ,„ = 0, since the 



motion is supposed to be symmetrical with respect to the axis, the above equation becomes 



d'W I dw ffp sin a 



+ - — + '^ = 0. 



dr r dr ix 



Let a be the radius of the pipe, and U the velocity of the fluid close to the surface; then, 

 integrating the above equation, and determining the abitrary constants by the conditions that w 

 shall be finite when r = 0, and iv = U when r = o, we have 



go sin a 

 IV = '^ (or - r) + U. 



SECTION II. 



Objections to Lagrange's proof of the theorem that j/" udx + vdy + wdz is an exact 

 differential at any one instant it is always so, the pressure being supposed equal 

 in all directiotis. Principles of M. Cauchys proof. A new proof of the theorem. 

 A physical interpretation of the circumstance of the above expression being an 

 exact differential. 



10. The proof of this theorem given by Lagrange depends on the legitimacy of supposing 

 M, « and w capable of expansion according to positive integral powers of t, for a sufficiently 

 small finite value of t. It is clear that the expansion cannot contain negative powers of t, since 

 «, V and w are supposed to be finite when t = Q\ but it may be objected to Lagrange's proof 

 that there are functions of t of which the expansion contains fractional powers of t., and that we do 

 not know but that ?<, u and w may be such functions. This objection has been considered by 

 Mr. I'ower*, who has shown that the theorem is true if we suppose ?«, v and w capable of 



expansion according to any powers of t. Still the proof remains unsatisfactory, in fact inconclusive, 



1 



for these are functions of t, (for instance e''' , t log /,) which do not admit of expansion according 



• Camltrulye l*h'tl(tsnj)hiral Trfiifuctionti, Vol, vil. Part 3 



Vol. VIII. Part III. R r 



