[SULLIVAN] 



A RULED SURFACE 



209 



0, 



which is the equation given by M. V. Jamet for a ruled surface (Nou- 

 velles Annales De Mathématiques, Vol. X, p. 500). 



A conclusion arrived at above provides a more direct method 

 than that employed by Jamet of demonstrating that an integral 

 surface of (B) is a ruled surface. In short, equation (B) expresses 

 the condition that the equations 



d~z_ 

 dy~ 

 dh . _„ d'z 



(9) 



^ +2X ^ +X^ 

 dx^ dxdy 



= 0, 



'^ +3X 



dx'^ dx^dy 



d^z 



dxdy"^ dy^ 



may have a common root. From the first of these we find by differenti- 

 ating with respect to x and y: 



d-z 



- +A — I 



dx 



(10) 



^+2X-^^ 

 dx^ dx-dy 



d^z 

 + X2-^ +2 



^_!i +2X ^ 

 ay dxdy- 



+x^ 



dxdy^ 



xdrav dv- / dx 



dx^dy \dxdy dy-y dy 



If cognizance be taken of the second of (9), the equations (10) show 

 that 



(11) Ci!^+x!2)('?^+xl^)=o. 



\dxdy dy-y \dx dyy 

 If the factor I +X — ) vanishes, the expression 



\dxdy dy-y 



Kd'z\ dH dH\ . ■ u ^ u 

 I — I must vanish and the correspondmg surraces 

 dxdy/ dx- dy- J 



are developable — a class not contemplated in (A). 



