Geometry on the Cubic Scroll of the First Kind. 53 



For the direction of the curve cp = at the points 

 where it meets the linear director we may proceed thus,— 

 Writing . 



the points where cp=r=0 meets the linear director are given 

 byUp_^^^ = 0; thus, if 



Up_ci = (ai>^ + \\^) (.S^^ 4- bofx) (ap_^/ + \_^^) = 0, 



then a^À -)- b^[j. = , agX -f- b2[x =:= etc. give the generators 

 passing through the points of the curve on the linear 

 director. 



V V 



Since Y = - =00 on the linear director, substituting 

 vj" )- for ^ ) [J^ 5 V respectively and clearing of fractions by 

 multiplying by X^^<i,^\'^^, we have 



^E^v'.U^^ + v'-'.U^.^ + v-'-lU^^.H- 



+ V . Up+q-1 + Up+d = 0, 



and our problem is now to find the tangent curves of 9 ^ 

 at the points where it meets the line - = -=0 (while - is 



^ X [X (J. 



finite), a problem corresponding precisely to that of finding 

 the asymptotes of a plane curve. 



Hence, if the point P' ^ (X' , ij, , 0) be an ordinary point 

 of ^ = 0, the tangent conic A'f' = gives its direction there; 

 if it be an 1-tuple point and l<q, the tangent 21-thic, and 

 if l>q, the tangent (1 -[- q)-thic will give the direction of 

 each of the 1 branches. 



