Section III, 1919 [207] Trans. R.S.C. 



On a Derivation of an Equation of a Ruled Surface. 

 By Charles T. Sullivan, Ph.D., D.Sc, F. R.S.C. 



(Read May Meeting, 1919). 



When a non-developable surface is referred to its asymptotic lines 

 as parametric curves, so that in terms of the variables u, v the equa- 

 tions of the asymptotic lines are « = const, and z; = const., the Gauss 

 equations of the surface take the form 



(A) 



(1) ^=r^+A^ 



du"^ du dv 



(2) ^=r"^+A"^ 



dv^ du dv 



(The symbols F, A, etc., used in these equations and the symbols A, 

 D, etc., used below are defined as in Forsyth's Differential Geometry, 

 pp. 45, 190, 192). If the asymptotic lines « = const, are plane curves, 

 they must be straight lines. In short, if the curves «< = const, lie in the 

 planes 



a{u)x-^b{u)y-{-c(u)z=l, 

 then 



ax2-\-by2-hcz2 = 0, 



(3) ax22+ by 22-\- CS22 = 



, dx dx d-X 



where Xi= — ■ , X2= — , Xoi= — , etc. 



du dv dv' 



These equations combined with {A —2) lead to the relation 



r"(axi + èji-fczi) = 0, 



The expression in brackets cannot vanish; because if it did, we should 

 conclude that 



axi-]rbyi-^czi = Q, 



(4) ax2-\-by2-\-cz<> = 0, 



axn-\-byi2-\-czn = Q, 



and, therefore, that MV=0. But this result is inconsistent with the 

 supposition on which the system (A) is based. We therefore infer that 

 r" must vanish. Now for the asymptotic lines the invariant A 

 vanishes and the invariant D vanishes along m = const., since T" 

 vanishes. Hence y = p = p = œ ; and the curve 7^ = const, is a straight 

 line. 



