Roberts— 0^/ soine Properties of a Certain Quintic Curve. 35 



If we now solve the above quadratic for the ratio z : y , it is plain 



we may write 



px = A6 



py = ^ _ \ (3; 



where H - Jb^-AC. 



It follows thus that the line x = dy meets the curve in two 

 points P and P', which we call corresponding points, and further, if 

 we denote their coordinates by x, y , %, and x\ y\ z', we shall have 



px = AO px' = A'e 



py = AT PI/' = '^ ^ • • (4) 



pz = £"+JP p%' = B-JT, ^ 



that is to say, 6 being given, to one point P there corresponds a 

 positive value of the radical J R , and^ to the other point P' a 

 negative value. It is evident that if ^ = 0, the points P and P' 

 coincide. 



2. — The class of this curve is easily ascertained, since the triple 



point is to be counted as equivalent to three double points in 



estimating the number of tangents which can be drawn from an 



arbitrary point to the curve ; this number is then 5x4-3x2 = 14; 



and if this point be on the curve the number of tangents which can 



be drawn from it will be 14 diminished by 2, or 12 ; but if the point 



be 0, the triple point, the number of tangents which can be drawn 



from it to the curve will be 14-3x2= 8: hence eight tangents can 



be drawn from to the curve. K"ow, a line drawn through meets 



the curve in corresponding points, and these points will coincide 



when the line touches the curve, the eight points of contact of 



tangents which can be drawn from to the curve, are then points 



which coincides with their corresponding points. Their equation is 



•consequently _ 



J2 = . 



"We shall call the roots of ^ = , aj , ao , . . . ag , and we shall 

 sometimes write it in the form 



J)i, D^, -Da, and D^ being four quadratic factors whose roots are 



