Geometry on the Cubic Scroll of the First Kind. 17 



If y' = 0, a' and |3' are both even in 1) and the change to 

 C'. z2 '^^ s 2 ß' follows, but oJ and {5' are both odd 

 in 2) and hence in general y'>1. 



If Y'=a' = 0,ß' is even and the change to C', s^ß' is at 



once possible. 



3) a'<;Y'<a' + ß' 



C'. X^V^'^"^' = C', x^'. yï' - '''. si(^' + ß ^ - Y'). 



It is now easy to see what the form of o) must be in 

 any case in order that the conditions be satisfied, which 

 are that 



p -f- n' be even, 

 _p + n'>2(q + n(), 

 I V be a factor of œ (unless œ . ü be a function of )C and [j."), 



and that n' be a minimum. 



All cases may be classified according as p is even or 

 odd and according to the relative values of p and q. We 

 have already seen that 



m' = Kp4-nO,f' = n^,g' + g^ + h' + h^ + l' + l^ = n'-n^', 



and . k' = i(P + iiO-(q + D(); 



and since here 



3m'-2f'-(g' + g^-l-h' + h^H-l' + l^)-k' = 



f(p + n') — 2n( — (n' — np — [l(p + n')-(qH-n()] = p + q, 



it is seen that in every case the entire residual intersection 

 is contained in 2f',g',gQ,h',h^,l',\' , and k'. 



Hence we may proceed at once to the following cases; — 



2 — Archiv for Math, og Nalurv. B. XXI. Nr. 3. 

 Trvkt den 14de Juni 189'). 



