SIR W. THOMSON ON VORTEX MOTION. 223 



and, by (9), 



cir 



where — denotes rate of variation per unit length perpendicular to the spherical 



dr 



surface that is differentiation with reference to r. the other two co-ordinates being 

 directional relatively to the centre. Hence, using ordinary polar co-ordinates, r, 

 6, y]s, we have 



w-'MZt**"** ••••• (»>■ 



But the " equation of continuity" for an incompressible liquid (being 



du dv dw _ 

 dx + dy dz "~ '* 



gives* y 2 <P =0, for every point within the spherical space; and therefore [Thom- 

 son & Tait, App. B] 



p = S + Sj r + S 8 r 2 + &c (12). 



a converging series, where S denotes a constant, and S a , S 2 , &c, surface harmo- 

 nics of the orders indicated. 

 Hence 



R = d £ = S 1 + 2r S 2 + 3 r 2 S, + &c. . . (13). 



And it is clear from the synthesis of the most general surface harmonic, by zonal, 



dS ■ 



sectional, and tesseral harmonics [Thomson & Tait, §781], that ^-£ is a surface 



harmonic of the same order as S ( :f from which [Thomson & Tait, App. B (16)], 

 it follows that, 



^2 ^2 ^2 



* By v 2 we shall always understand -^ + -^— 2 + -=-% . 



| This follows, of course, from the known analytical theorem that the operations v 2 and 



( y -j z -j- ) are commutative, which is proved thus : — 



By differentiation we have 



*(»2) 



d 2 dp d dtp _ 



J fill" fly fill fly 



dy % " dy 2 dz dy dz 



i ,, P . d dp d dip 



and theretore, since --- - - = - r 



dy dz dz dy 



^-*|)=^)--*(|M^-*I>'' 



or 



9 / d d\ C d d \ „ 



V [V dz- Z dy)? = (? dz ~ Z dy) V ?' 



p being any function whatever. Hence, if v 2 p = we have 



(y (J 1 _ g d l) = • 



V dz * dy) 



