SOLID GEUMKTIIY. 



VI. General Functional and Differential Equations. 



1188. A surface is generated by a straight line which always 

 intersects two fixed straight lines 



prove that the equation of the surface generated is of the form 



'mzx cy^ 



1189. The general functional equation of surfaces generated 

 by a straight line, intersecting the axis of s, and meeting the 

 plane of xy in the circle x* + y * = a f , is 



and the general differential equation is 



(* + f)(px + qy - *) = a f (px + qy)'. 



1190. The general functional equation of all surfaces, gene- 

 rated by a straight line which always intersects the axis of z, is 



and the differential equation is 



1191. The differential equation of a family of surfaces, such 

 that the perpendicular from the origin on the normal always lies 

 in thr plane of xy, is 



*(p* +?") + (/>* + ?y) = o. 



1 192. The differential equation of a family of purfaces, g 

 rated by a straight line which is always parallel to tin- )<luu> 

 of xy and whose intercept between the planes of zx, yz is constant 

 ;i:i<l equal to c, is 



