SIR W. THOMSON ON VORTEX MOTION. 231 



have for the two parts of this quantity considered above, and its whole value, the 

 following expressions; of which the first is taken in anticipation from § 42 — 



a> momentum of matter, within S, =ffNx da (8) of § 42 ] 



as-component of impulsive pressure on S, outwards, = — ff<p cos a. da J ^ '' 



X =f/Q$x - <p cos a) da ....... (2). 



It is worthy of remark that this expression holds for the impulse of all the solids 

 or vortices within S, even if there be others in the immediate neighbourhood out- 

 side : and that therefore its value must be zero if there be no solids or vortices 

 within S, and N and (p are due solely to those outside. 



38. If <f> be the potential of a magnet or group of magnets, some within S and 

 others outside it, and N the normal component magnetic force, at any point of S, 

 the preceding expression (2) is equal to the ^-component of the magnetic moment 

 of all the magnets within S, multiplied by 4^. For let p be the density of any 

 continuous distribution of positive and negative matter, having for potential, and 

 normal component force, <p and N respectively, at every point of S. We have 



[Thomson &Tait, § 491 (c)] s= - ^ v 2 p, and therefore 



JjTfivdxdydz = - ^fff x ^ai + i? + -J) clxd y dz - (3) - 



Now, integrating by parts,* as usual with such expressions, we have 



fjj^<tedydz=ffx g dy dz -JJf% dxdydz =ff(*% - f) dy dz . 



Hence, integrating each of the other two terms of (3) once simply, and reducing 

 as usual [Thomson & Tait, App. A (a)] to a surface integral, we have 



1 1 1 2 X dxdy dz = — -r- J J (Nx — <p cos a) da . . . (4) ; 



which proves the proposition, and also, of course, that if there be no matter 

 within S, the value of the second member is zero. 



39. Hence, considering the magnetic and hydrokinetic analogous systems 

 with the sole condition that at every point of some particular closed surface, the 

 magnetic potential is equal to the velocity potential, we conclude that 4^ times 

 the magnetic moment of all the magnetism within any surface, in the magnetic 

 system, is equal to the force-resultant of the impulse of the solids or vortices 

 within the corresponding surface in the hydrokinetic system ; and that the direc- 

 tions of the magnetic axis and of the force-resultant of the impulse are the same. 

 For the theory of magnetism, it is interesting to remark that indeterminate dis- 

 tributions of magnetism within the solids, or portions of fluid to which initiating 



* The process here described leads merely to the equation obtained by taking the last two equal 

 members of App. A (1) (Thomson & Tait) for the case a — 1, U = <p, U' = x. 



