266 Mr. Barlow on the rules and principles for 
If we make — = 9 : these equations are 
(« —y) r' 
^ -V qy 
(j —y) r 
1 + 
y 
= 2 m, 
= 2 n. 
{q -{• 1) <p X =z r* and — cp x — r. 
Substituting for r' and r in the two former, we have (ob¬ 
serving that ( 1 —jv) X =y, 
y (7 + I) 2 m 
* + V 9 ?» 
V (9 + o _ 
m , 
2 n 
=. n . 
q + y ' 9 
Hence m' q = y (^ + 1), 
and > = . 
, m' + (l — m) g'^ q 
9 +J' =— , 
And substituting the last two values in the equation pre 
ceding them, we have 
m* (a + i) , 
= n , 
m' -f (1 _ /„ ) ^ 
or, ml n' -{■ {n' — m' n') q^ n' q = m' {q 1 ). 
Whence (rrl n' — n') q^ (m' — rl) q = n* m! — m\ 
2 , m' — n' n m' — m‘ 
or, q H-r-;-> q = — -r~r -r- . 
And since here q — — 1 is obviously one of the roots, the 
= q, 
'll n — n ffl g \ 
— -^ _ _ _ (1.) 
Again, since ^ , and y = , , we may 
Other will be 
or q 
readily obtain 
2 m 
2 w) (q + I) 
- (2). 
