DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 367 
, , ™ //(/— 1) simp cosffl -v/I 
and therefore tanysecy=^ ~T^ ' LM (265.) 
Substituting this value in the preceding equation, vve get 
r y 
/ l-i^ 
Fdu 
V 1 yjLv/1 
) mvi ^ 
>■/(/- 1)J 
— V 
Jcosu" 
.... (266.) 
In (1/0.) we showed that, m and n being conjugate parameters connected by the 
equation m-\-n — 
.^1— dp dp i^rdp 1 Gdr 
V ^ JjN \/l' in JM V'T'” 
/I— n\ I /l—?\ /I— m\ I /■\—?\ ^ I , 1 I . / l—i^ 
( n ) i )’ \ m ) ^\l—\jmn Vmn 
m 
Substituting these values in the preceding equation, and dividing by we get 
If we add this equation to (266.), the coefficient of the integral will vanish, and 
the resulting equation will become 
VI 
‘dr ' 
ICOST 
(268.) 
fd? , rtip_ rdp VI rrdy r 
JLv/I^IN^I v/ (/-!)(/- i2)LJcosy Jc 
We shall now proceed to show thatj^^— may be put under the form 
if we make the assumption siny'=^^^^^^, (269.) 
V being equal to (1 — ??)^^— 1^ 
Now 
Hence 
„ (1 — m sin^p)(l — /sin^p) 
cos^y= as in (263.). 
cos^p 
I [1 —P sin^p — sin^p cos^p] 
LM 
/ / — ^ du Gdp Fi 
V /{/— l)lcoso Jv'Il 
but we derive from (165.) and (166.) the value 
/ I—? r dr _rdp [n cos^p— n sin^p + m^ sin'^p] 
V MN 
or subtracting. 
(270.) 
(271.) 
V /(/— l)l_JcOSO jcostJ 
^VT p 1 sin 
M LL 
N 
5~| , Gcos^p sin^p , n-\ . , ^ ^ 
-J'’?-Jm;7t[-L-+n]<'?- • • (272.) 
These two latter integrals may be combined into the single integral, 
'[1 — sin^p — n cos^p] [1 — sin'‘p]dp 
LMNv/r 
3 B 
f 
(273.) 
MDCCCLII. 
