68 



PROFESSOR CAYLEY ON POLYZOMAL CURVES. 



the rationalised equation is easily found to be 



— 2%z(l\a + w/x/3 + nvy) 



+ z\l 2 * 2 + m 2 /3 2 + n 2 y 2 - 2mn^y - 2nlya - 2lmap) = 0. 



And it is to be noticed that in the case of the circular cubic or when 

 J I + Jm + Jn = 0, then A = 0, so that the equation contains the factor *, and 

 throwing this out, the equation gives a single line, which is in fact the tangent of 

 the circular cubic. 



162. Returning to the bicircular quartic, we may seek for the condition in 

 order that the node may be a cusp : the required condition is obviously 



a(7V + ot 2 j8» + n 2 7 2 - 2mnfy - 2nlya - 2bna0) - (!ka + mp£ + my) 2 = 0, 

 or observing that 



A — X 2 = — 4wm, &c. 

 A + (iv = — 2/X, &c. 



this is 



/a 2 + m[3? + ny 2 + X/3y + pyx + raj3 = , 



or substituting for A, /*, v, their values, it is 



I (a - /3) (a - y) + m(fi - y) (|8 - a) + My - a) (y - /3) = , 



or as it is more simply written 



m 



+ — ■= = 0. 

 — y y — a a, — p 



163. If the node at (>i = 0, z = 0) be also a cusp, then we have in like manner 



P-7 



-, + 



m 



y — a 



ni-P 



= 0. 



Now observiDg that 



( y _ «) (a - &) - (y - a) (a - j8) , = 



a 



«', 



1 



|8, 



ft, 



1 



7' 



/ 



7> 



1 



= (a-/3) dS'-yO- (V-/3')(/3-y), 



= 08-7)(/--O- 03-/)(y-«). 

 = Q suppose : the two equations give 



I : m : n =s n(/3— y) (ft - y 7 ) : 11(7 - a) (7 - a') : n(a - .3) («' - /3') 

 or if Q is not = 0, then 



l:m:n = (p- 7 )(P-y):(y-a) (y - a') : (a- j8) (a'-ft). 



164. If 



n = 



a, 



a', 



1 



ft 



ft, 



1 



7> 



7, 



1 



= 



