Roberts — On some Properties of a Certain Quintic Curve. 39 



5. — To any point for whicli 6 is given, and also the sign of the 

 radical J^, correspond the three integrals Io{0), li{0), and l2{6), 

 which we shall refer to as the integrals of the point, to any three 

 points whose parameters are 6^, 62, and 6s, will correspond three 

 integrals Iq^Oi), Iq^Oz), and Iq{6z), three integrals of the class Ii{d) 

 and three of the class I^ifi)- If we write 





(1) 



we might, not improperly, call Wq, Ui, and C^, the arguments of 

 the three points whose parameters are 0^, 62, and 63. 



For such quintic curves we have a theory of residuation analogous 

 to Sylvester's TTieory of Residuation for the Cubic ; but as I have already 

 discussed such a theory in a Paper published in the "Proceedings of 

 London Mathematical Society" some years ago, and as the treatment 

 for the quintic is almost exactly similar to that I adopted in the case 

 of the uni-nodal quartic, I shall not do more than allude to it. 



We consider the equation of a curve of the mtt degree of the 

 form 



az - h = , 



which we call an curve, where a and b are binary forms of the 

 m — Itt and mth degrees respectively. Such a curve has for a 

 point of the m - Ith order ; and we find, by reasoning of a precisely 

 similar nature to that we employed in discussing the relations of the 

 parameters of the points of section of a line with the quintic, that 

 the 2m + 3 values of 6 corresponding to the 2m + 3 points of section 

 are connected by three relations, viz., 



^1{6) = iV. 



If two systems of points on the quintic a and ^ together make up 

 the complete intersection of an curve and the quintic, these 

 systems are said to be co-residual. 



6. — We now turn to the equation of the curve 

 Az^ - 2£% + C = 0. 



