228 SIR W. THOMSON OX VORTEX MOTION. 



in place of p (fdt denoting a time integral from any era of reckoning before the 

 disturbance became sensible, up to time t, which may be any instant during the 

 collision, or after it is finished), we have the changes in the corresponding com- 

 ponents of the impulse up to time t, provided there has been no impact of move- 

 able solid on the fixed solid. 



31. Let now the " velocity potential" (as we shall call it, in conformity with a 

 German usage which has been adopted by Helmholtz,) be denoted by (j> ; that is 

 (§ 16), let (/> be such a function of (x, y, z, t) that 



dp dp dp 



i' = ^-,v — -f-,iv=-f- . . . (3). 



dx ay dz v ' 



and let (p (or -^- j denote its rate of variation per unit of time at any instant t, 



for the point (#, y, z) regarded as fixed. 



Also, let q denote the resultant fluid velocity, so that 



2 2,2 2 "V , (1< P 2 , (] <P 2 fAS 



?2 = « 2 + * 2 + ^ = a£ + 3*, + £ • (4). 



The ordinary hydro-dynamical formula gives 



2> = n-p-£j 2 . . (5) ; 



where II denotes the constant pressure in all sensibly quiescent parts of the 

 fluid. 



32. The constant term II disappears from p in each of the integrals (1) of 

 § 30, because a solid is equilibrated by equal pressure around. And in the time 

 integral (2), we have 



fpdt = p . . . . . (6) ; 



and therefere if (XYZ) (LMN) denote the changes in the force-and couple-com- 

 ponents of the impulse produced by the collision up to time t, we have 



X= -Jfda cos a (p + 1/2 2 dt), Y = &c, Z = &c, j 



L = - ffdg (y cos 7 - z cos p) {p + $ f q 2 dfylHL = &c.,T$= &c, j ' ' ' ^ ' 



But because the fluid is quiescent in the neighbourhood of the fixed body when 

 the moving body or group of bodies is infinitely distant from it ; it follows that 

 before the commencement and after the end of the collision we have </> = at 

 every point of the surface of the fixed body. Hence, for every value of t represent- 

 ing a time after the completion of the collision, the preceding expressions become 



X = — \ffda cos u.f(f dt, Y = &c, Z = &c, \ 



L = - lffd<s{y cos 7 - z cos /3) fq 2 dt,K = &c, N = &c, J ' ' ^ ' 



which express that the integral change of impulse experienced by a body or group 

 of bodies, in passing beside a fixed body without striking it, may be regarded as a 



