PROFESSOR CAYLEY ON POLYZOMAL CURVES. 59 



which is ultimately adopted, we suppose that for the two nodes respectively 

 (£ = , z = 0) and (i, = , s = 0) , then if 1% + mz = , /'£ + m'z = , 

 nn + pz = , n'n 4- p'z = are the tangents at the two nodes respectively, the 

 equation will be 



(1% + mz) (l'% + m'z) (nn + pz) (n'n + p'z) + ez 2 %n + z s (a% + in) + cz i = 0, 



and if (in order to make this equation divisible by z, and the curve so to break 

 up into the line z = and a cubic) we write I = or n = 0, then the curve will 

 indeed break up as required, but we shall have, not the general cubic through 

 the two points (£ = 0,3 = 0), (n = , z = 0), but in each case a nodal cubic, 

 viz., if I = there will be a node at the point (n = , z = 0), and if n = a 

 node at the point (£ = , z = 0) . 



Analytical Theory for the Circular Cubic — Art. Nos. 142 to 144. 



142. I consider then the two cases separately ; and first the circular cubic. 

 The equation may be taken to be 



|«j(pg + qn) + cz%n + z 2 (a% + In + cz 3 ) = , 

 or what is the same thing 



%n(j>% + qn + ez) + z 2 (a% + In + cz) = , 



viz. (£, rj, z) being any co-ordinates whatever, this is the general equation of a 

 cubic passing through the points (£=0, 2 = 0) , (»7 = 0, z = 0), and at these points 

 touched by the lines £ = , n — respectively. And if (£, n, z = 1) be circular 

 co-ordinates, then we have the general equation of a circular cubic having the 

 lines £ = , n = for its asymptotes, or say the point £ = , n = for its 

 centre ; the equation of the remaining asymptote is evidently p% + qn + ez = ; 

 to make the curve real we must have (p, q) and («, b) conjugate imaginaries, 

 e and c real. 



143. Taking in any case the points /, J to be the points £ = , z = and 

 ^ = 0, z = respectively, for the equation of a tangent from /write p% = 6z; then 

 we have 



0j}(03 + qn + ez) + 2 (a & + bpn + cj?«) = , 

 that is 



z 2 (ad + cp) + nz(6 2 + e6 + bp) + n 2 .qd = , 



and the line will be a tangent if only 



{f- + c 9 + lp) 2 — 4qd (aO + cp) = , 



that is, the four tangents from I are the lines p^ — Qz^ where is any root of 

 this equation. Similarly the four tangents from J are the lines qn = <pz, where 

 <p is any root of the equation 



(p 2 + c<p + aq) 2 — 4p<p(b<p + cq) = , 



