ON POLYZOMAL CURVES. 
533 
414] 
156. Calculating AB'-A'B, GA'-C'A, CD'-CD, BD'-B'D, these are at once 
seen to divide by {(a/3' - a/3) H+ Z (a' - ¡3') -L'(a- /3)}; we have, moreover, 
BC - B'G = - 4 (a - /3) 2 (a'77 - L’) (/377 -L') + 4 (a' - /3') 2 (aTT - Z) (/337 - L), 
= - {(aa' - /3/3') H-L(a' -/3') - Z' (a - /3)} {a/3' - a'/3) 77 + Z (a' - /3')- Z' (a - /3)}, 
viz., this also contains the same factor ; and omitting it, the equation is found to be 
{(a - /3) (a' - /3') - 4 (/377 - Z) (/377 - Z') } 
— 2 {(aa' — /3/3') 77 — Z (a' — /3') — Z' (a — /3) } (AJ + A 2 ) 
+ {— (a — /3) (a' — /3') 4- 4 (aZT — Z) (a!H — Z')} AiA 2 = 0 ; 
viz., substituting for \i + A 2 and their values, this is 
{(a - /3) (a' - /3') - 4 (/37T - Z) (/377 - Z')} (717 + ZTaa' - La! - La) 
- {(aa'-/3/3')H-L (a'-/3')} {(a-/3)(a' -/3') + 43747-4ZZ'} 
+ {- (a - /3) (a' - /3') + 4 (aZT - Z) (a'ZT - Z')} {47 + 77/3/3' - Z/3' - Z'/3} = 0, 
which should be identically true. Multiplying by 77, and writing in the form 
{(a — /3) (a' — /3') — 4 (/377 — Z) (/377 — Z') } (3747-ZZ'-t-(aZ7- Z) (a'77-Z') ) 
- {(aTT - Z) (a77 - Z') - (/377-Z) (/377 - Z')} ((a - £)(«' - /3') + 4 (77717 - LL) ) 
+ {-(a- /3) (a' - /3') + 4 (aTT - Z) (a'77 - Z') } (7747 - ZZ' + (/377 - Z) (/3'77 - Z')) = 0, 
we at once see that this is so, and the theorem is thus proved, viz., that the equation 
being pp'qq' = V 2 , the foci (p = 0, p — 0) and (q = 0, q' = 0) are concyclic. 
157. By what precedes, A being a root of the foregoing quadric equation, we may 
write 
qq + 2AF 4- A 2 pp' = K 2 rr', 
where the focus r = 0, r’ = 0 is concyclic with the other two foci; but from the 
equation of the curve V = 'Jpp'qq, that is we have 
qq + 2A Vpp'qq + A 2 pp' = Krr’, 
or, what is the same thing, 
A \fpp' + Vqq -K y/rr = 0, 
viz., this is a form of the equation of the curve; substituting for p, p', q, q\ r, r' 
their values, writing also 
21 =(1;-*z){ V -a!z) ) 
%=(Z-/3z)( V -/3’z), 
® =d - V z ) (v - y' 2 )> 
.and changing the constants A, K (viz. A : 1 : K — Vi : Vra : Vw) the equation is 
( + V 5 + VW(5 = 0,