16 PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON, [489 
where, however, the II (x) of the last equation must be regarded as differing from that 
of the first equation by a period, say il, of the integral; hence adding, we have 
O = nil (k), 
or 
n (k) = - H, 
n 
which gives between the constants of the integral equation (*][#, l) 2 (y, l) 2 = 0, a 
relation which must be satisfied when the series closes at the n th term (viz. when 
the term after this coincides with the first term); and this relation is independent 
of x, that is, of the position of the point A. 
The analysis in regard to the differential equation is as follows: 
Consider the equation 
U = y 2 (ax 2 + 2 bx + c ) 
+ 2y (a'x 2 +2 b'x + c') 
+ (a"x 2 + 2 b"x + c") = 0, 
say 
U=(P, Q, R\y, 1) 2 = (£, M, N^x, l) 2 = 0, 
we have 
dU=0 = (Py + Q)dy + (Lx + M) dx. 
But the equation (7=0 gives (Py + Q) 2 = Q 2 — PR, (Lx + M) 2 = M 2 — NL, and the 
differential equation therefore becomes 
dy V(Q 2 - PR) ± dx s/(M 2 - NL) = 0, 
viz. it is 
dy 
\Z{(ay 2 + 2a'y + a") (cy 2 + 2c'y + c") — (bx 2 + 2b'y + b") 2 ) 
dx _ „ 
~ \/{(ax 2 + 2bx + c) (a!'x 2 + 2b"x + c") — (a'x 2 + 2b'x + c') 2 } 
Suppose the equation is 
y 2 (ax 2 + 2hx + g) 
+ 2 y (hx 2 + 2 bx + f) 
+ (gx 2 + 2fx + c) = 0, 
then the differential equation is 
dy 
V{(ay 2 + 2hy + g) (gy 2 + 2\fy + cf - (hy 2 + 2by +/) 2 } 
dx _ q 
“ \J{(ax 2 + 2hx + g) (gx 2 + 2fx + c) — (hx 2 + 2bx + f) 2 ) 
say