G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 268 
We shall be in want of the expression of SqII. Having 
T= Saea, 
we get 
ah = > 




Taking the scalar, the first term is its own scalar 
(Sda + Vda) (Sda — Vda) = S’<a — Vda. 
The second is, by change of place of a, etc., 
Sapa. 
Thus we get: 
(33) Sall=P—Q+SII’. 
Now, by (29) and (30) we have: 
1 2 ly > 
(34) Biz Qa, GIL] VW) 720. 
Therefore, and by (81), 
(35) SII = ; (SM — V°Il) + j30!. 
But we have interest to express S<II in P and Q alone, because these 
values will be immediate consequences of the data derived from the conditions 
of integrability. Therefore let us eliminate 2m,”’. By (30) and (31) we get 
02. sh: 0). 
Therefore : 
(36) Sal =,SM+P+Q. 
This formula shows also that 
(87) _ Sil’ + Sal = 5s. 
We may also express S<jII in relation to 2S [a<d(S<da)] in the following 
manner. Starting from (24), namely 
VII = 2208 <a, 
and applying the operator Sd, the first member becomes S<I], because 
<1(SII) being a vector, the scalar of it will disappear. Thus 
SdH = 228 [ aaSda 
+a (Sda) | 
The first term in the second member is 22(S<ja)? = 2P. Therefore 
VOL. XXVII. PART III. 3 Z 
