SIR W. THOMSON ON VORTEX MOTION. 221 



13. The following is Stokes' proof of Lemma (1) : — First, for any motion 

 whatever, whether subject to the condition of § 10 or not, let L be the component 

 moment of momentum round OX of an infinitesimal sphere with its centre at 0. 

 Denoting by fff integration through this space, we have 



L = ffj '(ivy — vz)dxdydz . . . (1). 



Now let (~ j , \~f-) , &c. denote the values at of the differential coefficients. 



We have, by Maclaurin's theorem, 



_ /dw\ /dw\ /dio\ 



w ~ x \dx) + V \dyj + Z \dz) ' 



and so for v. Hence, remembering that ( -j- ) , &c. are constants for the space 

 through which the integration is performed, we have 



fffdx dy dz wy = y~\ fffxy dx dy dz + f-^\ fffy 2 dx dy dz + (^J fffzy dx dy dz . 



The first and third of the triple integrals vanish, because every diameter of a 

 homogeneous sphere is a principal axis ; and if A denote moment of momentum 

 of the spherical volume round its centre, we have for the second 



(ffy 2 dxdydz = %A. 



Dealing similarly with vz in the expression for L, we find 



rdiv \ /dv \ 



1 -**[£).- ©J (2) - 



But L must be zero according to the condition of § 10; and, therefore, as the 

 centre of the infinitesimal sphere now considered may be taken at any point of 

 space through which this condition holds at any instant, we must have, through- 

 out that space, 



dw 

 dy 



dV rs 



dz 



du 



dw __ q 

 dx 



dz 



dv 

 dx 



du q 

 dy ~ 



(3); 



and similarly 



which proves Lemma (1.) 



14. To prove Lemma (2.), let 



dp dq> ds ,., 



U =dx> V =<Ty> W = dz ' ' ' (4); 



and let L denote the component moment of momentum round OX, through any 

 spherical space with in centre. We have [ (1) of § 13], 



VOL. XXV. PART I. 3 L 



