1896 - 97 .] Prof. Tait on liquations in Vector Functions. 505 
12. The most general linear equation in x and % may be written 
as 
Apply the members of the' given equation to a , /?, y separately ; 
and operate on each of the results with S.a, S ./?, S.y. We obtain 
nine scalar equations in x a = A, — /b X7 ~ v > which two are 
.<X(f)(q)\ -p q/Ji -p vv) ~p S.l/qa(s A -p t fJ. -p UV ) ) = Set Cq, , 
2(S /3</>(y?A + qp + Tv ) + S i/qa (s'X + t'\i + Uv ) ) = S/3£a . 
These are necessary, and sufficient, to determine A, /*, y ; and 
thence x- 
2 + ^xVi = £ • 
Take a, /?, y, three non-coplanar vectors, and let 
4> i a =_£>a + y/3 + ry n 
cf)^(3 = p'a +q'ft + r y V etc. 
if/' a — Sa + t/3 + Uy 
2 K 
VOL. XXI. 
