40 Proceedings of the Royal Iri8?t, Academy. 



If we write for z, Zi^f, /being the equation of a line tlirougli 0, 

 we find 



^Zi^ -2{B- 4/)si + C- 25/+ Af- = , 

 or, A%^ -2 B-^Zi -f C'l = , 



where B^^ B- Af, C, = C-2Bf+ Ap. 



Now it is clear since / contained but two constants, it will not in. 

 general be possible to make C\ contain the binary cubic ^ as a factor. 

 The curve we discuss is that special case of the quintic with a triple 

 point in which it is possible to transform the equation of the curve 

 so that Ci^ AQ, Q being a binary quadratic. This involves one 

 condition. The equation of this curve can then be written in 

 this form 



Az--2Bz ^ AQ = (1) 



This being the case, suppose we transform the above equation to a 

 new axis of z , which will be effected by writing 2 + / for z . For 

 this transformation we have 



Az^-2B,z+ C\ = 0, 

 where Bi = B - Af, 



C\ = AQ-2Bf+Af. 



Now, we say, since we have two constants at our disposal, that it is 

 possible to determine / so that B - Af may contain Q -f as a factor. 

 Let us write then 



B-Af^ {Q-f)F, (2) 



F being a bioary quadratic. 



Let us now see what C'l becomes in this case. We have 



C\ = A{Q+f-)-OBf 



= A{Q+f)-2f.[Af+{Q-f)F\ 

 = {Q-f){A-2fF) = {Q-r)Bse.j, 

 where E ^ A - 2f F . 



