¡7 
ie 
11' - 
fiämm 
667] 
ON THE BICIRCULAR QUARTIC. 
239 
drawn in the senses A to B, B to C, C to D, D to A: and representing the in 
clinations, measured from the positive infinity on the axis of x in the sense x to y, 
by v 1} v 2 , v 3 , v respectively: then, in passing to the consecutive quadrilateral A'B'C'D', 
we have and v 2 decreasing, v 3 and v increasing, that is, dv 3 and dv 2 negative, dv 3 
and dv positive; so that, reckoning the elements A A', BB\ CC', DD', that is, dS 1} dS 2 , 
dS 3 , dS, as each of them positive, we have 
dS 2 — dS 3 = — 28^!, 
dS 3 — dS 2 = — 28 2 dv 2 , 
dS — dS s = + 2 8 3 dv 3 , 
dS 3 + dS = + 28 dv , 
and thence 
dS = 8dv — 8 1 dv 1 — 8 2 dv 2 + 8 3 dv 3 , 
dS 3 = 8dv + 8 l dv 1 + 8 2 dv 2 — 8 3 dv 3 , 
dS 2 = 8dv — 8 x dv i + 8 2 dv 2 — 8 3 dv 3 , 
dS 3 = 8dv — Sjduj — 8 2 dv 2 — 8 3 dv 3 , 
which are the correct signs in regard to the particular figure. 
8 dco 
Reduction of |— —to Elliptic Integrals. Art. No. 29. 
of l< 
(x 2 + y-) V® 
29. The expression in question is 
dco. 
J(cos (w V/ +0 — a) 2 + (sin ft) ’s!g + 6 — ßf — 7 3 
V© 
lCOS" ft) Sin- col 
1/+0 
where V® is a mere constant; and we may apply it to the Gaussian transformation, 
a + a' cos T + a" sin T 
COS ft) = 
sm ft) = 
c + 0' cos T + c" sin T ’ 
b + b' cos T + b" sin T 
J ~~~ IP 1 v" IP > 
c + c' cos T + c" sin T 
where the coefficients a, b, c, a, b\ c, a", b", c" are such that identically 
1 
COS 2 ft) + sin 2 ft) — 1 = 
and also 
(c + d cos T + c" sin T) 
(cos w */f+ 6 — a) 2 + (sin co V# •+■ 6 - ß) 2 — 7 2 , 
cos 2 T + sin 2 T — 1}: