56 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
2 
Now the equation of the base of the hyperbolic cylinder being 
let 
,r=asec0, ?/=&tan0, .... 
then 
tan \L=-=- sin#, 
and 
d* b nln a 2 
co^-a coM9 ’ cos ; 
hence 
j , ab cos ddd 
a 2 + b 2 sin 2 # 
Since 
, r 2 a 2 + b 2 sin 2 # 
tan ^—£ 2 — A 2 cos 2 # 5 
g 2 4- A 2 cos 2 # 4- b 2 sin 2 # 
SeC ^“ A 2 cos 2 # ; 
[A 2 cos 2 # 4- a 2 4- b 2 sin 2 #] 5 
. sec f/j A 3 cos 3 # 
Let (AT) denote the area of the logarithmic hyperbola, then 
3 ( AT) = k 2 ^ 
’[A 2 cos 2 # + g 2 4- A 2 sin 2 #] 2 ab cos ddd 
A 3 cos 3 # [a 2 + b 2 sin 2 #] 
Let \ = k 2 cos 2 0+a 2 +6 2 sin 2 #, 
and the last equation will become 
■k 2 tan ‘^sin#^ . . . 
3(AT) = f T 
L 
abk 2 cos 2 ddd . C2abkdd abC \a 2 + b 2 sin 2 #] 
I [a 2 + b 2 sin 2 #] y/y~^J 
and this may be written in the form 
3(AT) = 
Let 
ak 3 (a 2 +b 2 ) 
dd 
a 2 + b 2 sin 2 #) y/y ' 
dd 
*1 A 2 tan“'(- sin^) 
7 , gA 3 g# 3 
2 cibk — ^ p 
dd 
y/y 
+j(a*+4*)J ; 
COS 2 # y/y' 
■ k 2 tan 1 ( - sin#^ 
A 2 
= t.an 2 g=w. 
A 2 -6 2 
k 2 + a 2 '' 
. ; 2 
and the preceding equation may be written 
3(AT) = JV±£> I' M 
' ab \/a 2 + k 2 J [1 + n sin 2 #] vT —i 2 sin 2 # 
+ 
ab(a 2 + b 2 ) 
f. 
! J< 
d6 
(k 2 —b 2 ) 
2 r% 
k \d a 2 4- A 2 J cos 2 # v 7 1 — i 2 sin 2 # bk y/ « 2 + k 2 ] \/ 1 — i 2 sin 2 # 
## 
-Ac 2 tan Y-sin#^) 
Since 
w=-s 
A 2 1 + n a 2 + b 2 
~—~W~ ’ 
,25 
k 2 
• /c 
and as (1 — m)(l +w)= 1 — « 2 , and (47-) gives 
/ I 4- w \ d£ /I — m\ f 1 dd i 2 Cdd 1_ ^ ^ wm sin# cos#\ 
\ n /Jn vT \ m /Jm vT _ ™nj Vl' Vma a ° 
a/I 
(446.) 
(447) 
(448.) 
(449.) 
. ( 450 .) 
