386 Mr. Brougham's general Theorems 
1 : 9 : P “ 1 “ an( * * m + n : : q*—p : m -f- 2 n f AR is 
equal to AG x q _ __ . anc j } because LM is a conic hyperbola, the 
p + 
m -f n x q — p 
m + 2» 
rectangle MS . RS, or MS . AP, or AP . MP -j- AR is equal to 
the parameter, (or constant space,) therefore, this parameter 
is equal to AP x MP + AG . q 
P + 
(m + n) (q — p) 
m -f 2 n 
Again, the space ACD is equal to ------- of the rectangle 
AC . CD, since AD is a parabola of the order m + n ; but (by 
construction) AC . CD is equal to 
m -f n 
of Q 
M-f N 
~M 
, AN; 
therefore, ACD = 0 
M + N 
M 
. t . AN, of which 0 : parameter 
of LM : : tt ; M, and vr : M -f- N :: <p -|- p ; q — p\ therefore. 
Par, lm x m + n + p-\ also, r : q : : AG : 
M {q — p) l m 4 - 2 n 1 ’ * 
0 
m + 2 n 
q — p; consequently, ACD — — ^ multiplied by 
m + " + p) and diminished by x AN x — ■ 
m -j- 2 n 
q-p 
therefore, transposing 
equal to ACD -J- 
Par. LM x M + N j m + n q — p 
M 
m x q — p 
X AIM X ‘ 
m-f 2 n . 
X 
P J is 
M + N AN x — — ; and Par. LM will be 
acd + x an x ^7-^— j x 
M 
equal to - 
J!L ± J L x r± + P ) x mTn 
m - f 2 22 / 
- — — — - — -, that is. 
M 
M + N 
X q — p X ACD -f q • AN x AG 
m n 
m -f 2 n 
X q —p + P 
