206 Mr Brill, The Application of Matrices to the [April 30, 



Further, we have 



dw = {du~-^dv^) e^^'^^'^'^.Fil v) 



= du . e 



where ^ and rj have to be made to vanish after the operations have 

 been performed. Thus we shall have an equation of the form 



dw = du. U+dv . V, 



where U and V are formed from dF/d^ and dF/drj in the same 

 manner as tu is formed from F {^, 77). 



4. It can be proved that the sum of all the products involving 

 u and V respectively r and s times can be expressed in the form 



k^ + k^u + k^v, 



where k^, k^, k^ are scalars; or, substituting for u and v their 

 values, in the form 



ki f hp + keq. 



Further, the matrical coefficients. A, B, G, &c., can each be 

 expressed in the form 



^1 + kV + h^l + hP^l' 



where l^, l^, l^, l^ are scalars. Thus our function w may be ex- 

 pressed in the form 



T + Xp + Yq^-Zpq, 



where X, Y, Z, T are scalar functions of x, y, z. Hence, if we 

 write 



d^ .d^ a^ . d' d"" , d^ 



we have 



VT + ^X.p + VY.q + VZ.pq = 0, 



from which we may conclude that 



VT = VZ = VF=V^=0, 



provided that the determinant 



0, 13„ /3„ «A + AS2 



0, 7i> 72' 7i«2 + ^i72 



1, ^1. ^2' 7A+M2 



