Geometry on the Cubic Scroll of the First Kind. 15 



identically vanishiog expression; therefore we can always 

 put fo==0. The tangent plane to 2 along the linear director 

 is given by x^s — y^z = , and that k^>0 the lowest terms 

 in z,s in the equation of S must contain x^s — y^z which 

 as just seen will never appear in S from the performance 

 in to . cp of the necessary substitution ; consequently we can 

 always put k^ = 0. The tangent plane to 1 along the pinch 

 point generator À = being z = 0, in order that go > it 

 is only necessary that the lowest terms in x , z in the equation 

 of S have z as a factor; this will occur if the terms lowest 

 in Àv,X" in o) . cp, as grouped for the substitution, all contain 

 X , which may be the case. Likewise the tangent plane to 

 2 along the other pinch point generator |jl = being s = 0, 

 in order that h^ > it is only necessary that the lowest 

 term in y,s in the equation of S have s as a factor; this 

 will occur if the terms lowest in |j.v , jj," in u> . cp, as grouped 

 for the substitution, all contain [x", which may be the case. 



Similarly for 1^> , 1^' > , 



Since in general X and \i may be interchanged in «>, we 

 may reduce (o to the zeroth degree in either X or [j- without 

 affecting the order m' of the surface S. Hence it will in 

 general be possible to reduce either g' -{- g^ or h' -|- h^ to 

 zero, a reduction in either being accompanied by an increase 

 by the same amount in the other. And since in general 

 either À or jjl or both may be changed for a corresponding 

 power of aX -|- bji- or to an arbitrary polynomial of the same 

 degree in X and [x, it will generally be possible to reduce 

 both g' + gg and h' -f- h^ to zero while 1' or 1' + 1^ is- 

 increased by the same amount. Exceptions to these 

 general statements will be noted later. Since f comes solely 

 from the introduction of v in o) and since (o is always 

 chosen of the minimum degree and no reduction by any 



