208 THE ROYAL SOCIETY OF CANADA 



Let X, y, z be three independent solutions of (A), and let us effect 

 the following transformation on {A): 



x=x{u, v), y = y{n, v), z=9(u, v). 



The relations between the two sets of variables and their derivatives 

 of the first and second orders are: 



^^ .,1 ^^ ^1 



— =xip+yig, — =X2p-'ry2q, 



dii dv 



(5) 



— =x\r-\-2xiyiS+y-it-\'Xnp-\-ynq, 

 du- 



=XiX2r-\- {xiy2-\-X2y\)s-\ryiy4-\-Xnp-\-ynq, 



dudv 



rfif) 



— =x~2r-\-2x<>y2S-\-y\t-\-X'>2p-{-y22q, 

 dv~ 



dz dz dH dh , dh 

 where p= — , q= — , r— — , 5 = , / = — . 



dx dy dx" dxdy dy- 



If these values be substituted in (A), we find the relations: 



.^x x\r+2xiyis+y\t=0, 



x\r-\-2x2y2S-\-y"2t= 0. 



On differentiating the second of these with respect to v there results 

 the relation: 



(7) dx^ dx-dy dxdy- dy 



+ 2{X2X22r + X2yi2+X22y2-S-\-y2y22t) = 0. 



Now for a ruled surface in which 7^ = const, are the generators, T" 

 vanishes. Hence in virtue of equations (5) and (A-2) : 



{x2X'iâr-\-X'iy22-\-X22y2-s-\-y2yi2t) = Ù," {x\r-\-2x2y2S-\-yy) = 0. 

 Thus for a ruled surface we have the equations : 



dx" dxdy dy- 



(8) 



dx^ dx^dy dxdy^ dy^ 



where \ 



\x->y 



