Geometry on the Cu))ic Scroll of the First Kind. 28 



generator or any combination of any number of arbitrary genera- 

 tors amounting all together to 2q — p --j- 1. In general here we 

 may have g^ -f- ^o + K ^ ^ ' '^"*' ^^'^^^ always g' + h' -|- 1' 

 must be > g,^ + h^^; -[- 1^'^, it follows that here g' + h' -j- 1' > 

 i(2q — p -|- !)• Since a surface of order m' can contain no 

 line of multipicity >> m' — 1 , it is seen that when a single 

 generator 1' counts 2q — p + 1 times in the residual intersec- 

 tion we must have either ln>0 or else q<p — 1 so that 

 2q — p -j- 1 ^ m' — 1 ; if in this case p = q , then must 1^ > 1. 

 Again, if Uj,= Vj,_ot. V,^, we may iput m = ^^^^^^^_^^. 

 ^Vot where V'^„^ is what V ,^^^ becomes when \i is changed 

 to — [J- in it, and o),,^^ t_- o ^® ^^ arbitrary polynomial in X" 

 and [x" of degree 2 (q -{- 1 — p) . Then will m' == p — t , f =0, 

 g' + g^ + h'-fh^ + l'+l^ = p-2t,k' = p-q-t,if t< 

 p — q, and otherwise, — 



III', p < 2q , t > p — q , m' = q , f ' = , 



g^ + g;; + h^ + h^ + l^ + l^ = 2q-p,k'^0. 



Hence for every curve p < 2q we can find S with f = , 

 andif t>p — q this S will have m' a minimum, to . cp has a 

 2)air or pairs of branches at every point where (p meets the 

 double director; at 2t of these points both branches belong- 

 to cp, at the remaining p — 2t points one branch belongs to ç 

 while the other is a generator belonging to a>. [cf. Note, p. 25]. 



Describing S and S' thus, — 



|S;m'=:q,f'=:0,g'-|-g^ + h'-fh^-fl' fl^=2q-p,k'=--0, 

 S';m" = q4-l,f" = l,g"-|-g^'-fh" + h--fl"4-l^'=: 

 i 2q — p + l,k" = 0, 



where g" + g^' -f h" -f- h;;' -f 1" + ],'/ refers to generators of 

 which none lie on S and where t > p — q ; or with m' = p — 1 



