SIR WILLIAM THOMSON ON VORTEX MOTION. 239 



■-Jf*i~ >%-$//*?-¥ ■ ■ ■ ^ 



and (19) becomes 



'-(^•TT • • • ■ <*>' 



Whence, if X denote the ^-momentum of the fluid at any instant in the space 

 between concentric spherical surfaces of radius r and /, 



O r 3 _ ,,'3 



x -i-r bi .... (23> 



If r and r' be each infinitely small in comparison with a, this expression vanishes, 

 as it ought to do, in accordance with § 45, II. But if 



- = , & r = a , ) 



( . . (24), 



it becomes X = — § . 4«-A ) 



fulfilling § 4, by showing in the fluid outside the spherical surface of radius ?-' a 

 momentum equal and opposite to that (§ 45, II.) of the whole matter, whether 

 fluid or solid, within that surface. 



53. Comparing § 47 and § 52, we see that if X, Y, Z be rectangular com- 

 ponents of the force-resultant of the impulse, the term T a r~ 2 of the harmonic 

 expansion (14) is as follows : — 



T .^Xa+Yy + Zs (25) 



provided all the solids and vortices taken into account are within a spherical 

 surface whose radius is very small in comparison with the distances of all other 

 vortices or moving solids, and with the shortest distance to the fixed bounding 

 surface. 



54. Helmholtz, in his splendid paper on Vortex Motion, has made the very 

 important remark, that a certain fundamental theorem of Green's, which has 

 been used to demonstrate the determinateness of solutions in hydrokinetics, is 

 subject to exception when the functions involved have multiple values. This calls 

 for a serious correction and extension of elementary hydrokinetic theory, to 

 which I now proceed. 



55. In the general theorem (1) of Thomson & Tait, App. A let a = 1. It 

 becomes 



JIf(%dl + %Ty + %^^y^=JJd^M'-JJJdxdydz 9 ^' 



— I j ihf'iip — IJI dxdtjd~<pv'-$ .... (1), 



which is true without exception if (p and <p' denote any two single- valued fxmvtiows 

 of x,y,z; Jffdx dy dz integration through the space enclosed by any finite closed 



