PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



61 



and r) = is the harmonic of the same line z — in regard to the tangents at 

 (rj = 0, z = 0). If (£, f], % = 1) be circular co-ordinates, then we have the general 

 equation of the bicircular quartic having the lines £ + az = 0, £ — as = for 

 one pair, and the lines >? — /3s = 0, >7 4- /3s = for the other pair of parallel 

 asymptotes; and therefore the point £ = 0, n = for centre, and the lines 

 /3£ — ay] — 0, /3£ + a?? = for nodal axes. In order that the curve may be real 

 we must have (a, /3), (a, J) conjugate imaginaries, #, 0, c real. The points 

 (£ = 0, z = 0) and (17 = 0, = 0) are as before the points 7, J. If a — 0, the 

 node at / becomes a cusp, and so if /3 = 0, the node at J becomes a cusp ; the 

 form thus includes the case of a bicuspidal or Cartesian curve. 



146. To find the tangents from 7, writing in the equation of the curve £ = 6az, 

 we have 



la 2 (6 2 - 1)0 2 - jSV) + eatvs + z(aata + h) + cz 2 = ; 



that is 



n 2 . k* 2 (P - 1), 



+ %z . ea$ + b , 



+ z 2 . - £a 2 j6 2 (0 2 - 1) + aad + c = , 



and the condition of tangency is 



4k(& 2 - l){/,-a 2 /3 2 (0 2 - 1) - aaO-c] + (cO + -) = ; 



viz., the tangents from / are £ = 6az, where S is any root of this equation. 

 Similarly, if we have 



4k (f - 1) {/v« 2 /3 2 O- 1) - JjSp - c] + (e<p + ~y= , 



the tangents from J are y = <fij3z, where <p is any root of this equation. 



147. The two equations may be written 



24&V/3 2 , 

 - 6M/3, 



24&V/3 2 , 



— 6kaa , 



- 8/vV/3 2 - 4kc + e 2 



6 



aa 



+ 3e - 

 a ' 



b UU) 4 =<M 



- 8Fa 2 /3 2 -4ftc + e 2 



24&V/3 2 + 24&c + 6 



6 2 



6A*/3 



+ 3e 



)> (ft 1)* = , 



24Pa 2 jS 2 + 24&c +6^ 2 



which equations have the same invariants ; in fact for the first equation the 

 invariants are found to be as follows, viz., if for shortness 



then 



C = - 8k 2 a 2 j3 2 - 4kc + e 2 , 

 I = 576&*a 4 /3* + 576Pca 2 /3 2 + 144F(a 2 a 2 + b 2 /3 2 ) + 72kab + SC 2 



J= C{576^ 4 a 4 /S* + 576Fca 2 /3 2 + 144& 2 (aV + 5 2 /3 2 ) + 36&e«jS - C 2 } 

 - S64khaba, 2 ^ 2 - 216 Jc 2 e 2 (a? a? + b 2 P 2 ) - 216 k 2 a 2 b 2 , 



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



VOL. XXV. PART I. Q 



