252 



Double Generation. 



The law of double generation is simply stated. Two straight lines are 

 chosen parallel to the plane of the curvilinear directrix, the three giving 

 rise to a scroll of a certain equation. Suppose two other straight lines can 

 now be found parallel to tlie plane of the curvilinear directrix and inter- 

 secting the first two rectilinear directrices. Suppose the use of the second 



pair of lines gives exactly the same equation as the first two, then the sur- 



px 

 face is one of double generation. For example, x^ y'^c. Substitute -^ — 



qy 

 for x^ making p ^ 1 and for y^ making q = — 1. There results 



q — z 



xy 



^= c; now make p = — 1 and q = + 1. Tlie same equa- 



(1 -f z)(l-z) 



tion results. In fact these are the two generations of the hyperboloid of 



one sheet. 



It then becomes at once apparent that all scrolls are doubly generated 

 whose curvilinear directrix has for its equation a function of the product 

 term (xy), the plane of the curvilinear directrix being parallel to tlie recti- 

 linear directrices. Thus the first of the five 4th scrolls order mentioned 

 above, viz. : the one having x^y^ and c, and perhaps x y terms in the 

 equation of the curvilinear directrix is a scoU of double generation. 



It is not at once evident that the property discussed above is co- 

 extensive with all the doubly generated warped surfaces in the family 

 under discussion. Such surfaces may also depend upon other properties 

 not yet discovered. 



General Observations. 



It is evident that the validity of the demonstration does not require 

 the axis of Z to be the common perpendicular between the two recti- 

 linear directrices. If the Z axis connects the two directrices in ques- 

 tion and passes through the middle point of their common perpendicu- 

 lar, it follows at once that the demonstration proceeds as before l)y 

 parallel instead of orthogonal projection. 



If we conceive tlie three axes of reference, under the restrictions just 

 given, to be oblique to each other, we find the resuUing equations are still 

 in their simplest forms. In the surfaces of the second order the axes 

 would then be conjugate axes. In surfaces of higher order the axes 

 of reference would play the part of conjugate axes. 



