492 DR G. PLARR ON THE DETERMINATION OF 



If in the second terms we permute the letters one step forward in the series 

 a, b, c, a, &c., the terms become respectively 



— Ea 2 5 2 5 1 & 2 a 1 2 c 2 c 3 

 + 1,a 2 b 2 b 1 b 3 a 1 2 c 2 c 3 . 



Joining them to the first terms we have 



yuY' = "Ea^a^c^c^b^ — a^] 

 zuZ' — 2a 2 6 2 a 1 & 1 c 2 c 3 ( — b^ + afy]. 



And as 



b x a., — a^,, = S(3aj i = SyJc = — c 3 ; 

 — b 1 a 3 + a 1 b 3 = Saftk i — Syj — — c 2 , 



we get 



t/hY' = — Ea^a-p^^c^ = — Y a , 



zuZ' = — 'Za 2 b 2 a l b l c 2 2 c 3 = ~-Z x . 

 We may remark that the second members may be put under the form 



+ ~Za 2 b 2 [a 2 b 2 + a 3 b 3 ]c 2 c 3 2 (in the case of the first equation) 



= aW[w 2 'E| 2 +W 3 'Sf 3 ] 



so that 



yuY' = a 2 Z> 2 c 2 [W 2 'S£0& + W 3 'Sj<f>k] . 



Likewise in the case of the second equation 



zuZ' = a 2 b 2 c 2 I,[W 2 'Sk<pj + W 3 'S?0j]. 

 Hence dividing the first by W 2 ', the second by W 3 ', and remarking that 



Y' v? Z' u s 

 we get 



W 2 ' v ' W 3 ' w 



= ^a*b 2 c 2 [Sk<pk+^,Sj<t>k] 

 , = 5aW[S/0y+^S%-]. 



<fe_W 2 ' 



As 



a§rw; 



we may also write these equations under the form 



yW 2 'dt w = vS.dp<f>k, 

 zW 3 'dt iy = wS.dp(j>j 



dt lr being a convenient infinitesimal scalar. 



