C 513 ] 
Cafe II. Given p, m, n : Required r. 
Since r 4- il — 
1 P 
I 
Therefore r — — 1 . 
P 1 
Cafe III. Given a y m y n : Required r. 
Since A — 1 x a = mr. 
Therefore ('i=H =) - = IZELzi. 
\ nr ' na nr 
Now r+rl = i -\-nr\n x - — ?r*4-»x- — ! X- — 
2 23 
&c. 
Therefore — == i 4- ” — L r 4- - — I x - — - See. 
na 2 1 2 3 
2 2 
« j rn r~i ~ , n — i ! « — i . n — 2 
And — =i + r J x r | ; 
an\ 2 2 3 ! 
which, by the Binomial Theorem, will become 
= 1 + r + ’pp r% (nearly). 
Let ° = (i"’ =) 1 
*n a\ 
12 
12 
Then ¥-r— D— i x Let 2 E:=-FL > 
«+ 1 »+i »+i 
Therefore r = F — E. 
2d, find E 
- I 
+ 1 
2 
m 
n — 1 
na 
n + i‘ 
Vol. LX. 
3d, find F = V2 x L> ~ 1 +E x E, 
4th, find r — F — E. 
CJuu Cafe IV. 
