G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 259 
y 1 
: a-— 5 Van; 
5 , ih 
(21) B, a 5 V Ba, 
, 1 ; 
i 5 Vyo: . 
Let us with these values of a; etc., calculate the expression of VI, namely, of 
the vector of I. 
We have by definition (as generally 


Vir= Ss 



aiey | > 
Developing Vaaa;,, we get for it 
aSaa, + a,Saa, 
the negative term being zero because of its factor Saa, = 0. 
Substituting now (21) for'a,, etc., and inverting the order of summation : 
aBy 
apy 
abe 
aera 
Vib= z2 > 




fa S(aVaw.) + Vaw,Saa .} 
In the first term we have: 
SaVaw, = Sag,a = SaVaa,: 
Thus by (A) § 1 we have 
il 
—52 

aBy | as(aVo.a) = + > Vo.e- 
Likewise the second term under the form 
—3v{> aBy 

aSaa } a, = 4 : Vazw,. 

Therefore we have 
(22) Vi= : > vey V(o.a + aa.) ; 


a result which is identically zero, because (7,4 + aza,) is a scalar, 
Thus we have 
Therefore I = Sa<a is a scalar. 

| By (16) we have for this scalar 
SI =— SII. 
We recapitulate now these results of reduction. We have by the preceding : 
VOL. XXVII. PART ITI. 3 Y 
