92 CORRECTION OF TWO THEOREMS RELATING TO THE PORISM, &C. [ 11 6 
The second theorem gives the condition in the case where the conics are replaced 
by the circles x 2 + y 2 — R 2 = 0 and (x — of + y 2 — r 2 = 0, the discriminant being in this case 
— (1 + £) {r 2 + £ (r 2 + R 2 — a 2 ) + g 2 R 2 }. 
As a very simple example, suppose that the circles are concentric, or assume 
a = 0; the square root of the discriminant is here 
(1 + f) Vr 2 + ; 
JR? 
and putting for shortness — = a, we may write 
A + B£ + ... = (1 + £) Vl + a£, 
that is, A= 1, B = ^a + 1, C = — ±a 2 + %a 2 , D = ^ a 3 -±a 2 , E = - a 4 + -faa 3 , &c.; 
thus in the case of the pentagon, 
GE — D 2 = JT fo « 4 {(« - 4) (5a - 8) - 4 (a - 2) 2 } 
== ToV? 0,4 ( a “ — 12a + 16), 
and the required condition therefore is 
a 2 -12a+16 = 0. 
It is clear that, in the case in question, 
¿=cos 36° = i(V5 + l), 
that is, ^ = Vs — 1, or (R + r) 2 — or 2 — 0, 
viz. (Va + l) 2 — 5 = 0, or a + 2 Va — 4 = 0, 
the rational form of which is 
a 2 — 12a + 16 = 0, 
and we have thus a verification of the theorem for this particular case. 
2 Stone Buildings, Oct. 10, 1853.