160 THE ROYAL SOCIETY OF CANADA 



From these relations and equation (12), it follows by the direct 

 differentiation of (15) that 



ôlog , — ôlog , — 



du du ' dv dv 



Thus the point tt is fixed in space; and, therefore, the locus D of the 

 directrices (8', ô") is either a quadric cone or a plane pencil. 



Equations (14) shew that the tangent PyPz to T'(v) at the point 

 Py meets the flecnode curve on R" in the point 



aiog- 



(16) f = ^ -^+,; 



o u 

 and, in the same way, the corresponding point on R' is 



a log ^=^ 



(16)' f' = ._-^ + p. 



av 



Equations (15) and (16) readily lead to the equations 

 / — 51og , — 



(17) 



a log , — , — 



Therefore the plane P^PirP^' is tangent to the loci of Pf and Pf' at 

 the points f and f' respectively. 



We shall now prove that the plane P^PxP^' is fixed in space; 

 and that, therefore, the locus D \s a plane pencil. To prove this it 

 will be sufficient to shew that each of the functions f, f' satisfies a 

 completely integrable system of three partial differential equations of 

 the second order; because such a system has precisely three linearly 

 independent solutions.* These systems will be found by differentiat- 

 ing (17) with respect to w and v, and eliminating in turn the variables 

 (f , tt) and (f', tt) from the resulting equations. The equations satisfied 

 by f are found to be 



*N, pp. 474, 475. 



