176 



THE ROYAL SOCIETY OF CANADA 



But it was assumed at the outset that ^i, 62, 63 were independent 

 solutions of (A). The only tenable assumption is. then that the locus 

 of (v], V2, V3) is a straight line. This assumption is also sufficient to 

 insure the vanishing of F". 



We may assume vi, p-i, u-i to have the following values: 

 Vi = aip{q)-\-bu v-2 = a2p(q) + b2, v-i = a3p(q)-\-h, 

 where a-i, bi are constants and p (q) is an arbitrary function. The 

 equations of the surface can now be written 



(13) 



x = 



y= 



2= 



(a2M3) + (Ms) 



(a3Mi) + (Mi) 



(aim) + (61^2) 



P+(62M3) + 



p+(63Mi) + 



p+(èiM2) + 



iliiduz) , 



inzdfii), 



(mi^M2) • 



V da da^y 



Let us now proceed to the general case when no assumption is 

 made as to the range of the solutions of (A). As before the condition 

 to be satisfied is the vanishing of T", and this condition implies the 

 equation 



002 dWz" 



dq dq^. 

 It therefore follows that relations of the form 



(14) adi -\-bd2 +63 =0, 



a — +0 — H =U, 



dq dq dq 



d%.,d%.d% . 

 a — -\-b — + — =0 

 dq"^ dq^ dq^ 



must exist, where, as yet, a and b are not known to be other than 



functions of p and q. 



From (A) we find 



(15) j;;i_^d\^_^^dd^ 



dpdq^ dq dq 



From (14), by differentiating with respect to p and taking cog- 

 nizance of (14) and (A), we find 

 (16) da ddi d^ ^^^ = 



dp dq dp dq 

 Again from the same equations and (15), we infer 



O I — i/i -r A f-rci — 1/2 -TA f -r I 



^dq dq 



\dq dq/ \dq 



\dp dq- dp dq^y 



dq 



) 



