258 G. PLARR ON THE ELIMINATION OF, a B, y, ETC. 
Let us consider II under its explicit form (15). We group the terms by 
columns, and put by definition : 
(17) w, = aa, + BB + yy: 
for any scalar variable ¢; in this present instance ¢ represents a, 8, c. 
Thus we have : 
(18) Il=> 


The terms which compose a; are vectors, because the differentiation of 
v=—1,P=—1,7 =—1, 
in respect to any scalar variable ¢, gives 
Saa;, = 10, SB; =' 0, Syy, =O; 
so that aa; , BB; , yy; , are their own vectors. 
As we have III =— KII, we get 
II =— S2'o,4 + V2'u.d ; 
But Va,0 =— Vag, , because a, is a vector; thus: 
a 
aD, « 
(19) Li > 


We remark, that we have generally 
" (20) —m =+ (aa+ BB + v7). 
But one must not be tempted into taking a, as an exact differential quotient. 
We express a; , 8; , y; by the help of a,, as follows :— 
Multiplying, as for example, by a, both members of (17), and znto a, both 
members of (20), and adding we get : 
ap, — wa = aa’ + aBB’ + ayy’ 
+aa™> + B’Ba + y'ya. 
We group the terms of the second member into three groups, which give: 
ata’ + aa? =—2a’ 
ab’ os yyo = ye’ fs 7B —— a 
ayy’ + B’'Ba =— By — By =—a@. 
The first member is 2Vaw,, and the second member becomes —4a,. _ Operat- 
ing in the same way with B and with y, we get the formulas : 

