The collected mathematical papers of Arthur Cayley.

532

ON POLYZOMAL CURVES.

[414

153. We have to show that the four foci (p = 0, p'=0), (q = 0, q'= 0), (r = 0,
r'=0), (s=0, s'=0) are a set of concyclic foci; that is, that the lines p=O, q=O,
r = 0, s = 0 correspond homographically to the lines p' = 0, q' = 0, r' = 0, s' = 0; or, what
is the same thing, that we have
1, a, a', aa' =0,
1, /', /3/'
1, l, '/
1, a, ', 'at
or, as it will be convenient to write this equation,
a-/3 y-8 _ -a- /3-Y
a'-^ ' '-' - P'-8' '-
154. We have
/3 + 2X1L + X12a, ' + 2XL' + X12a'
7= 1 +2HX2+X12' = l+2HX,+X2
/ + 2X2L + X2C, ' + 22L' + X22a'
1 + 2HX2+ X2' 1 + 2HX2+ X22
The expressions of a-, &c., are severally fractions, the denominators of which disappear
from the equation; the numerators are
for a -, = a ( -+ 2X2H + H2) - (/3 + 2X2L + aX22),
= a-3 + 2X2 (aH-L);
for /3 -, = 1/ (1 + 2XH + Xi2)- (/3 + 2X1L + aXI2),
= {2 (3H - L) (a-/3)};
for y - 8, = (/ + 2LXi + aX12) (1 + 2HX2 + X2)
- (f + 2LX2 + aX22) (1 + 2HX1 + X12),
= (a' - ') {2H2a-2L 2 ( +3) 2 + 2 (a - )2};
and it hence easily appears that the equation to be verified is
2H2a/3 - 2HL (a +3) + 2L2 + - (a - _)2 a - / + 2 (aH - L ) X 2(/3H-L)-(a -/3)X1
2H2a//3'-2HL'(a'+/3')+ 2L'2+ (a' /3 /)2 a' - /' + 2 (a'H- L') x 2 (1'H - L') - (a'- /3') X2
155. This is
B-C A + BX1 + CX2 + DX1X2
B'- ' -A' + B'X1 + C'x + D'X1X2
if for shortness
A= 2(a-/3)(3H-L), A'= 2(a'- 3') (/3'H-L')
B =- (a - 3)2, ' =- (a'-t _ ')2
C = 4 (aI - L) (/H-L), C' = 4(a'H- ') ('H-L'),
D=-2(a-/3)(aH-L), D =-2 (a' - ') (a'H- ')
and the equation then is
AB' - A'B + CA' - C'A - (x + x,) (BC' - B'C) + x1x, (CD'- C'D - (BD' - B'D)).