60 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



writing the two equations under the forms 



r 6 > 





r 6 > 





Be, 



- (6, l) 4 = 0, - 



3e, 





e 2 + 2bp - 4aq, 



c 2 + 2«q — 4zbp, 



•(M) 4 = o, 



3 cbp — 6 cpq, 





3eaq —Zcpqt 





- 6iy, 





- 6a 2 q 2 , 





the equations have the same invariants ; viz., for the first equation the invariants 

 are easily found to be 



/ = ?,(c 2 -4bp- ±aq) 2 + 72(«s - 2ab)pq, 



J= - (e 2 - 4:bp - 4aq) z - 36 (cc - 2ab)pq(< 2 - Up - ±aq) - 21(i cy-'i 2 , 

 and then by symmetry the other equation has the same invariants. The 

 absolute invariant P+ J 2 has therefore the same value in the two equations; 

 that is, the equations are linearly transformable the one into the other, which is 

 the before-mentioned theorem that the two pencils are nomographic. 



144. The two equations will be satisfied by = </>, if only bp = aq; that is, if 



p = 7 , q — t ; putting for convenience ' in place of e, the equation of the curve 



is then 



gu 0£ + bn + ez) + kz 2 (a£ + bn + cz) = . 



In this case the pencils of tangents are a£ = kOz, by = kBz, where 9 is deter- 

 mined by a quartic equation, or taking the corresponding lines (which by their 

 intersections determine the foci A, B, C, D) to be (a£ = k9 x z t b>i = k9 x z), &a, these 

 four points lie in the line a^ — bn = , which is a line through the centre of the 

 curve, or point £ = 0, n = : the formula) just obtained belong therefore to the 

 symmetrical case of the circular cubic. Passing to rectangular co-ordinates, writing 

 z — 1, and taking y = for the equation of the axis, it is easy to see that the 

 equation may be written 



(a 2 + y 2 )(x -a) + Jc(x-b) = 0; 

 or, changing the origin and constants, 



xy 1 + (x — a)(x — b) (x — c) = . 



Analytical Tlicory for the Bicircular Quartic — Art. Nos. 145 to 149. 



145. The equation for the bicircular quartic may be taken to be 

 fc(g 2 - a 2 z 2 )(n 2 - |8V) + cz 2 & + z*(a% + bn) + cz* = , 

 viz. (£, >7, s) being any co-ordinates whatever, this is the equation of a quartic 

 curve having a node at each of the points (£ = 0, z = 0) and (>? = 0, z = 0) : the 

 equations of the two tangents at the one node are £ — az = 0, £ + az — ; and 

 those of the two tangents at the other node are n — $z = 0, n + fiz = ; £• = 

 is thus the harmonic of the line z = in regard to the tangents at (£ = 0, z = 0), 



