68 Transactions of the Royal Canadian Institute 



k = ~ , where o- is the radius of torison of the complex curve at the 



point. 



If now the parametric curves g = const, on the surface (S) belong to 

 the complexes given by equation (11), and K be the Gauss Measure of 

 Curvature at the point P on (S), then it follows from Enneper's theorem 

 and equation (14) that 



1 



(15) — = V-K = S{a-\-bz-cy)^. 



Again from equations (7), (9) and (15), we deduce 



a. ft. 



(16) X = —=- , Y = 



V(rg VcTg Vo-g 



where X, Y, Z are the direction cosines of the normal at the point 

 P {x, y, z) on (S). 



Thus 



6i = V(Tq X = a-{- bz — cy , 



(17) $2= Vcq Y= 13-hcx-az, 



63 = VcTq Z = y + ay — bx. 



These are the equations we require to demonstrate that equation (A) 

 is of range three at most. We must first recall however the steps of 

 a theorem due to Goursat : 



Given a linear partial differential equation of the second order 



(B) -^ +a{p,q)-^ + b{p,g) — -\-c{p,g) = 0, 



dpdq dp dq 



if (n+1) linearly independent intetrals are connected by a homogeneous 

 linear relation the coefficients of which are functions of one of the inde- 

 pendent variables only, then n — 1 applications at most of a Laplace 

 transformation will lead to an equation with a vanishing invariant. 

 First consider the case n=l; then the assumed relation is (say) : 



02= U{q)di. 



If this value of 62 be substituted in (B) we shall find: 



dp 



