G. PLARR ON THE SOLUTION OF THE EQUATION Vpop=0 ° 83 
Retaining the definition of a, B, y, as a system of treble rectangular unit 
vectors, we put 
a = (ga, B= (P—-9)B, n= (¥—g)y, 
=(¢-g)a, Bi=(9—-9)B, ni = ($-9)y, 
and we define | and wW’ by 
ya = VB » VWB= Vyi0 » by = Va, , 
Wa=VBiyn, WB= Vyia, Wy = VaB,, 
the root g being supposed the same in both. 
Representing by w any vector, putting 
wo = au + Bv + yw, 
we have generally 
| pD = Yo = ua + whB + why 
= Wo = ua + wy'B + wyy, 
D and D’ being scalars whose values are to be determined in the sequel. We 
know that the unit vectors of yo, w/w are invariable; we will establish now 
that the tensors of these expressions depend on the direction of »o. 
Let us suppose 
ya = Ap, YB=Bp, py =Cp 
wa _ A’ p's WB _— B’p’, w'y — Co’. 
A, B, &c., being particular values of D D’. 
We find a relation between these scalars by the following way, namely :— 
Any scalar of the product of three vectors, a,, 6,, y, suppose, satisfies to 
the identity : 
38a,By1 = a1VBy + BiVyia + yi:1VuB,, 
and a similar one written for a;6;7; . 
But Sa,Bry, and Sa,Byy; are both equal to zero, as they represent /g, for the 
three roots of 7g = 0. 
Then we replace VB: , &c., by their expressions ya, namely mug &ce., 
and thus we get the two equations : 
0 = (A + BB + y,C)p 
0 = (a,A’+ BB+ ¥:C)p’, | 
and as the tensors of p and p’ are not generally vanishing, we have the. 
equations : 
0 = Aa, + BB, + Cy, 
0 = A’a, + BB, + Cy’ 


