234 
ON THE BICmCULAR QUARTIC. 
[667 
and similarly 
= a x x 2 + Ay* -(/+ 0 2 ) x,x 2 - (g + 0,) y$ 2) Vil 2 = - a,x, - A2/1 + (/+ 0 X ) x x x„ + (g + 0 X ) y x y 2 
Vn 2 = a^3 + &y, - (/+ 0 3 ) xa, ~(g + 0 3 ) y 2 y„ Vil : , = - a : Jc 2 - (3,y 2 + (/+ (9 2 ) ¿yr 3 + (y + 0 a ) y 2 y 3 , 
VIT3 = a 3 # + fty -(/+0)**» - (y + 0>M» = - a x 3 - /3 y 3 + (/+ 0 3 ) + (g + 0 3 )y 3 y. 
Differentiating the equation 
(/3 - A) (« - a?0 - (a - a a ) (y - y x ) + (0 - 0 X ) (xy x - x x y) = 0, 
we have 
[(/3 - A) + (0 - 0i) 2/1] d# - [(a - a0 + (0 - A) ^1] dy 
- [(/3 - A) + (0 - ^i)y ] ¿«1 + [(a - a,) + (0 - 0 X ) x ] dy x = 0; 
and writing herein 
dx = — ^ y dw, dx x = —-- — 2/i dcoj, 
V© * V©: * 
we find 
dw 
V© 
( ^r )a ' d< "> d *~ {l ^y x ' da ' 
{(g + 0)(/3- A) V + (/ + 0 ) (« - «0 ® + (# - A) ((/+ + (y + 0 ) Mi)} 
+ ^={(^+ A) (/3 - A) 2A + (/+ A)(a - «0 + (0 ~ 0i) ((/+ 0i) ocx 1 + (y + A) yyO} = 0 ; 
viz., dividing by 0 — 0i, this becomes 
dw deoi -, . . cfow , dw x _ 
— VD -7= — Vil = 0, that is, -=_ = 0 ; 
2l V© V ©j V© Vil VQjVili 
or, completing the system, we have 
dw _ — dw Y _ dw 2 — dw, 
V© Vil V©! Viîj V© 2 Vü 2 V'QgVOg’ 
which are the differential relations between the parameters w, w x , w 2 , w 3 , or (x, y), 
(x 1} y a ), (x 2 , y 2 ), («3, 2/s). 
22. From the equations X = a + -R'#, F = /3 + 72'y, we found 
dZ= J%i F -to+w. 
dF = {*-(/+*)«}; 
the new values, JT = otj + R x x x and F= A + A2/1 > give in like manner 
dX =-7im, {Y - (g+Wi ' 
dY =-^ym} x - (f+e ' )x ' ] -'