Roberts — On some Properties of a Certain Quintic Curve. 41 



The equation of our curve can consequently be written 



A%^-<l{Q-f)Fz + {Q-p)H= 0. . . (3) 



Now the equation of the tangents from to this curve will be the 

 discriminant of the above equation considered as a quadratic in z . 



Hence E ^ {Q-P)[{Q-p)F - AE} = AAA A; 



we have, consequently, 



A being the equation of a pair of tangents from to the curve 

 multiplied by such a factor so that 



There are eight R points as we have stated, and there are con- 

 sequently twenty-eight ways of arranging the eight points in pairs ; 

 and consequently twenty-eight ways of reducing the equation of the 

 curve to the form 



Az" - iBFz-^ BH = (4) 



If we write A = aiO^az, di, a-^, and a^ being the tangents at 

 triple point, we can show that our quintic can be regarded as the 

 envelope of a certain cubic curve. Let h be so chosen that 



Tcaitt'i + B = (fi^ say, 



then we have, multiplying the equation (4) by k and substituting 

 for kazttz its value (fy^ - B, 



ffi(<^2- A^' - 2BFkz + JcBE = 0; 



or, fl!i (<^2 - By = B{a,z^ + 2z (JcF- a^4>) + a^B - kM} 



showing that the curve is the envelope of the cubic 



aiB{p-iy - 2(p - l)(kF- <^«i) 



+ [a^z^^ 2z{kF-a,<j>) + aiB-kE'\ = 0. 



or, aiz'^ + 2z{kF- aip(f>) + p%B-kBr = 0; . . . (5) 



and this cubic touches the curve in five points where it meets 

 the conic 



z<fi = B p . 



