270 G. PLARR ON THE ELIMINATION OF a, 8, y, ETC. 
By the preceding formula (56) the first term becomes — p(ppq), thus we 
have 
> (peg) = KL 4(<p.9)]. 
If we limit the application of this to taking its scalar, we get 
Ce) «8p. b%q) = 8(4"p.9), 
because the terms 


d 
oe 7 bg.i = S4ppg 
ord 
S215 47 |i<p a, =Sdprg 


disappear, being common to both members. The result (57), we own it, may 
be easily guessed, in representing p and q respectively by w + #0, w—a, ete. 
However, we remark that the two results (56) and (57) take place inde- 
pendently of the assumption that the tensor of the quaternions Tp or Tq is 
constant. 
But now we establish this condition, by definition, and differentiate pg = 1, 
and Tp = 1. This gives severally 
(58) Past PG = 0, Sprig = 0; 
Differentiating a second time in respect to the same variable ¢, we get 
(59) Did + 2 + PI = 9; 
where we remark that the condition Tg = constant renders the second members 
equal to zero in (58) as well as in (59). 
Writing (59) successively for ¢ =a, =6, =c, and summing, we get the 
result | 
—Vp.g + 22|abe| pga —pbo*g = 90. 
As the scalars of the first and third terms are equal, we draw from this and 
from (57) the result 
(60) Ss 
dp 
abe === =) 4179.9 = Spe7g- 


The first term may be expressed by the derivatives in respect to a, b, ¢, or by 
those in respect to a, y, ; m all cases we have 
dp 
ee Pe 
> 

ml 


The demonstration of this may be conducted in the same way as the trans- 
formation in § 1 of formulas (1) into formulas (10 07s). 
We now have recourse to the intermediary of a; , etc., in order to form the 
relation between ,7, pq;, and a. 

