DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 57 
hence 
(v 2 ) 
(Hr) ' / '«" = 
dd i 2 pdd 
A =\" 77 + tan -1 
But 
Hence 
M \/ 1 V mnj V I 
k(a 2 + b 2 ) 
V mn sin# cos#~] 
VI 
J. (451.) 
3 (AT) 
ab 
ab V or + k 2 
dd 
k 2 
'k Vk 2 + a 2 ' 
k 2 
,?+5* sil ^] v'l 
ab{k 2 -b 2 ) f## fl% 2 + 6 2 ) C dd 
+ F \/F+^J \/F+?Jc 
+ tan -1 
COS 2 # 1 
V mn sin# cos# 
— tan -1 !""- sin# 
L VI J 
(452.) 
Now if Y be an arc of the plane hyperbola of which Vk 2 —b 2 is the transverse axis, 
and i the reciprocal of the eccentricity, we shall have 
C dd 
2 J 0 
aby ab[a 2 + b 2 ) 
A 3 fc 3 Va 2 + k 2 lcos 2 # VI 
(453.) 
And if we take the spherical ellipse whose principal semiangles, a and (3, are given 
by the equations 
b b fk 2 -\- a 2 
COSGJ = £, COS/3 = £\/ /( .2 + £2, 
we shall have 
and 
also 
sin 2 : 
tan/3 
k 2 -b 2 , k 2 
ab 
: k 2 + a 2 ’ e k 2 + a 2 
. COSCi = - -—ry 
tanst kVk 2 + a z 
•4/=tan -1 ^ sin#^. 
Hence the sum of the first and last terms may be written 
r , 
tan/3 p 
dd 
1 v - 
L_ 
“ a COS&1 _ 
tana 1 [1 - 
- e 2 sin 2 #] Vl — sin 2 s sin# 
and this expression is S, the value of the area of the spherical ellipse (a/3), as shown 
at page 16 of the Theory of Elliptic Integrals, 8$c. 
Now, as before, A being the transverse axe of the auxiliary hyperbola, 
A= V]p—b 2 , and B= Va 2 -{-b 2 , 
’ dd „ ... ab A 2 . 
f du ab Af 
may be written p ^ j, and the equation (452.) finally 
assumes the form 
3/f(AT)=«&r y+^J-^]-& 2 S+F tan- 1 
V mn sin# cos# J 
VI 
(454.) 
Or the area of the logarithmic hyperbola maybe expressed as a sum of the arcs of 
MDCCCLIV. I 
