DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 55 
Therefore, integrating the preceding expression, 
dd 
3 ( AH ) «^p+0 2 j[l-e 2 si 
sin 2 0] VT — « 2 sin 2 0 
2abk 
X—=M, 
2 j V 1 — i 2 
sin 2 # 
(444.) 
Vk 2 + a]} 
+ v' a 2 -\-k' 2 ^dd ^ \ — i 2 sin 2 6—k 2 tan~’^tan#) 
Hence the area of the logarithmic ellipse, or rather the area of the paraboloid 
bounded by the logarithmic ellipse, may be expressed as a sum of elliptic integrals of 
the first, second and third orders, with a circular arc. 
(I & — ^2 — ^2 ^ 
Since ~ a 2 e<i > i 2 > or the function of the third order is of the circular form. 
Assume a spherical conic section such that 
a b d* — b 2 
tan «=£, tan p=j., i = ; 
’a 2 + A 2 ’ 
therefore 
tan/3 
tana 
cos u— 
bk 
a V a 2 + k 2 ’ 
Sin-g= 
-// 
cd-b 2 
cd-\-k~' > 
Combining the first and last terms of the preceding equation, they become 
■ ¥ 
, (b r\ tan/3 C 
tan -tan# — - — cosal- 
\a J tana J[l — e 2 si 
d9 
~l 
sin 2 #] s / 1 — sin 2 e sin 2 # 
Now this is the expression for the surface of a segment of a spherical ellipse whose 
principal angles are 2a and 2j3*. Let this be S. 
In the next place, k s/ a 2 -\-k 2 ^dd v'' 1 — i 2 sin 2 # 
is a portion of the elliptic cylinder whose altitude is k, and the semiaxes of whose 
base are VV+A 2 and ^li 2 -\-k 2 . Let this be E, 
^ abk dd 
an V d 2 + k 2 j VT —i 2 sin 2 # 
is an expression for an arc of the spherical parabola whose focal distance is one-half 
the focal distance of the former. Let this be denoted by P. 
Hence if we denote the entire surface round Z by [AH], 
3[AH] = 4AE+-^pP-4**S (445.) 
Or the area of the logarithmic ellipse may be expressed as a sum of the arcs of a 
plane ellipse, of a spherical ellipse, and of a spherical parabola, multiplied by constant 
linear coefficients. 
LXXXV. To find the area of the logarithmic hyperbola. 
k 2 C 
The general expression for the area, as in (437-), is ^-Usec 3 ^— 1)^. 
* See the Theory of Elliptic Integrals, and the Properties of Surfaces of the Second Order, applied to the 
investigation of the motion of a body round a fixed point, p. 16. 
