88 UNIVERSITY OF COLORADO STUDIES 



x= 



where m and w, can assume the values and 1. This gives for the 

 arguments of the points of tangency of the four tangents from a point 

 u of the cubic 



2"' 2~' 2~' 2 (^^) 



Ti. , /. . p . „ . 2mw-\-2m.w, ^, ^ _ 



if we take for u a point oi inflexion ^ — -, tne arguments or 



3 



the points of tangency become 



2mw-\-2iVjWi {7n-{-^)2w-\-2m,^Wi 2mw-{-[m^-{-B)2w^ 

 - 6 '~ 6 ' 6 ' 



(m+3)2w?+(77?.,+ 3)2«j, 



6 



(22) 



4. The theorem contained in formula (18) may be generalized. 



A curve C„=0 of the nth. order cuts the cubic in 3/?- points. 

 Sn — 1 points may be chosen arbitrarily, and these determine the 

 remaining point. To prove this it is to be remembered that the 



. r. ^ . (^+lU^+2) , . ^ . . 



equation U^ — contains -^^ '-^ arbitrary coetiicients in a 



lii 



linear and homogeneous combination. The 371- points of intersection 



are not changed if the curve C„=0 is replaced by a curve with the 



equation. 



C'„ = C„-/(^,y)C„_3=0, 



where C„_3 is a polynomial of degree n — 3. As this polynomial 

 contains ^ '—^ arbitrary coefficients it is possible to dispose 



of these in such a manner that ^^ '-^ ^ terms in C'„ disappear, 



BO that only 



(/.+ !) {n^-2) _ {n-2) {n-l) ^^^ 



