Geometry on the Cubic Scroll of the First Kind. 21 



If it be desired to decrease k', while m' and f are 

 increased, we may put to^v^'^+i giving 



m' = i(p + l) + T,f' = 2x+l,g' + h' + P=:0, 



where x has the values , 1, 2 , , |(p — 1) — q. 



Again, if Up = Vp_2t . V^^ ,we may put co = V'^ g^ where 

 ^'p-2t ^^ what Vp_2t becomes when [i, is changed to — jx in it. 

 In this case the aggregate œ . U^ = V'^_,^. Y^_^^ • V,^ = f (}?, ^i^) , 

 the conditions of p. 17 are satisfied and we have 



n'=p — 2t,n^ = 0, 

 giving 



m' = p-t,f' = 0,g' + g^ + h'+h^ + l'+l^=:p-2t, 



k' = p — q — t 



for any integral value of t from to i (p — 1), (save that 

 m' may not be minimum). Thus every point of œ . cp = 

 on the double director has a jJaïr or pairs of branches passing 

 through it, one branch of the pair lying in either sheet; at 

 2t of those points both branches of each pair belong to œ, 

 at the remaining p — 2t points one branch of the pair belongs 

 to cp, the other branch is a generator and belongs to to. Since 

 p is here odd, o> must contain at least one generator to be paired 

 with the odd branch of cp , and consequently in no case will 

 a curve of this class, p > 2q and odd, be the complete inter- 

 section of S and 2. In general here g^ = h^ = 1^ = and 

 no change in the residual intersection is possible without 

 increasing m'. 



If t take its largest value, t = | (p — 1), we have 



i^'=4(p + i),f'=o,g'-i-h' + i'=i,k' = Kp + i)-qj 



