Geometry ou the Cubic Scroll of the First Kind. 25 



The formulae above give for the curves (p,q) arranged 

 according to their orders the following characteristics of the 

 surfaces S of lowest orders m', which cut those curves 

 from 2 , for all values of p and q save q = ; — 



[Note ; In III' we have given all the generators as unchangeable 

 in the residual intersection; it is evident, p. 23, that when t>p — q 

 an arbitrary polynomial a)o(q_|.t_p) enters; hence 2 [t — (p — q)] genera- 

 tors, which occur in pairs in the residual intersection, may be chosen 

 arbitrarily, save that they must occur in pairs, one branch of each 

 pair lying in either sheet; the remaining p — 2t generators are un- 

 changeably fixed by the curve.] 



