262 G. PLARR ON THE ELIMINATION OF a, B, y, ETC. 
We come to the expressions II’ and II” defined in (14), Such combina- 
tions, where a would follow >’a, or b(<a), or where <a or 4(fa) would 
enter, may be easily deduced from II’ and II”, so we consider only these two, 
and we confine ourselves to the valuation of their scalars. 
da 




We have, remembering a, = EE? ete., 
D>?a =— (dia + a, + ay). 
Therefore | 
SIV =—Z\aBy|> Be NEG, « 
If we differentiate a? =— 1, 8’ =, etc., twice over in respect to a, or 6, or 
c, we get 
 Saa, = 0, ete. 
and 
Saaja + G, =O, eto, 
Thus : 
SII’ =+ S|aBy|S|abe|a? 
Interverting the order of summations, and replacing aj, etc., by their expres- 
sions (21), we get SII’ to contain 
1 
iz 

Des 
a By | Via, = {> 
because w,, etc., are vectors, and by (E), § 1 
Therefore we have 
(31) SIV = : > 

EOC Nase. 

For the evaluation of SII”, we proceed from the formula 
P=] 04a, 
and remark that I is a scalar = — SII (22 07s), and therefore <I will be a 
vector, and consequently S<I will be zero. Thus we get 
a fy |Sdada +> bee S ia Sb 
In the first term we have 
=e 


S(<a)? = 97 4a + Vea. 
In the second term, being under the sign scalar, we shift a to the third place. 
Thus we get 
0= la By |{S?da + V?da + Sab (<a)}, 
and from this we deduce 
(32) Si” =e oe 

