266 Mr. Barlow on the rules and principles for 
if we make — ~ q : these equations are 
(i — y) r' 
~ * + qy 
0 — y) r 
1 + JL 
= 2 m, 
= 2 ft. 
( q + i ) <p x = r' and <$ x — r. 
Substituting for r' and r in the two former, we have (ob- 
serving that ( i — y ) * =y> 
V ( 7 4- i ) 2 m 
s + yq 
y (<] + O 
2 // 
— m' 
= n . 
•i + y - <p 
Hence ml + my <7=y(q+i), 
and J' = F=VT7+-F . 
„ 1 v — m ' + + 9 
H 8^ J (1 — m ) q 4- 1 
And substituting the last two values in the equation pre~ 
ceding them, we have 
m ' (<7 + 0 n ! 
m‘ + (1 — //, ) q* + q * 
or, m' n' - j- ( n' — m' n ' ) q x -f- n! q = ml ( q -j- 1 ). 
Whence (ml n" — n ' ) q 2 + ( m' — n' ) q =, n“ m! — ml , 
„ . wz' — re' 
or, q H ; — ; > q 
5 i 1 in 11 — re “ 
re m — wr 
»»' re' — re' 
And since here q — ■ — 1 is obviously one of the roots, the 
<1> 
■ - - ( 1 .) 
other will be m " ~ " 
or q 
rej re — «' 
2 m n — n <p 
2 mu — re <p 
Again, since y = — ~ , and y 
7 TT J— > We ma T 
(1 — m ) q + t ’ J 
readily obtain 
2 m 
2 m) (q + s) 
- (>)• 
