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 = 0, q = 0, 
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', 
0L0L ' 
1, 
¡3, 
/3', № 
1, 
7 > 
7> 
77 
1, 
8, 
8', 
88' 
or, as it will be convenient to write this equation, 
a —/3 7 — 3 _ a — 8 /3 — 7 
154. We have 
yS + 2X 2 Z + V« / y8 7 + 2A]Z 7 + A^a 7 
7== T+SIaW 7= 1 + 2H\ l + V ’ 
j /3 + 2X 2 Z + X 2 2 a /3 + 2XoZ + X 2 2 a 7 
= 1 + 2m., + V ’ = 1 + 2ZTX 2 + X 2 2 * 
The expressions of a —8, &c., are severally fractions, the denominators of which disappear 
from the equation; the numerators are 
for a - 8, = a (1 + 2X 2 H + X 2 2 ) — (/3 + 2X 2 Z 4- aX 2 2 ), 
= a — /3 + 2X 2 (aZT — Z); 
for ¡3 — 7, = /3(1 + 2X x Zf + Xj 2 ) — (/3 + 2XjZ + aXd), 
= Í 2 (/5^ — X) (a — ; 
for 7 - 8, = (¡3 + 2ZXj + aX x 2 ) (1 + 2H\, + X 2 2 ) 
— (/3 + 2ZX 2 + aX 2 2 ) (1 + 2-flAj + Xj 2 ), 
= («' - /3') {2ZT 2 a/3 - 2HL (a + /3) + 2Z 2 + * (a - /3) 2 } ; 
and it hence easily appears that the equation to be verified is 
2ZZa/3-2ZfZ(a + /3) + 2Z 2 + i(a-/3) 2 a -/3 + 2 (aZT - L )X 3 2 (/3 ZZ — Z ) - (a - ¡3)\ 
2H 2 a'{3'-2ZTZ 7 (a 7 +/3 7 ) + 2Z' 2 + ¿ (a' - /37 ~ a 7 - /3' + 2 (a 7 ZT - Z 7 ) X 2 ' 2 (/3 7 ZT - Z') - (a' - /3') X, * 
155. This is 
Z — C A + B\ l + (TX 2 + ZXjX 2 
Z^C 7 = 2 ' + Z 7 X 2 + 0% + DXX, ’ 
if for shortness 
A= 2 (a — /3)(AH — L) , ¿' = 2(a'-/3')(/3'ZT-Z') , 
Z=- (a — /3) 2 , Z 7 =- (a 7 — /3') 2 
C' = 4 (aZf - Z) (/3ZT- Z), <7 = 4 (a'ZT - Z 7 ) (/3 7 ZZ - Z 7 ), 
D = — 2(a — /3) (aH — L) , Z> 7 = -2(a 7 -/3 7 )(a 7 ZZ-Z 7 ) , 
and the equation then is 
¿Z 7 - ¿'Z + GA' - G'A - (X, + A,) (.BC - B'G) + X 1 X 2 (CD' - G'D - (BE - ED)).