54 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
or performing - the integration with respect to (x, 
area=^rjflfy sec constant. 
Now when the area is 0, sec (*=1, and therefore 
constant = — — Whence 
area=j^%//(sec 3 i w— 1) (437-) 
This is the general expression for the surface of a paraboloid between two prin- 
cipal planes, and bounded by a curve. 
When this curve is the logarithmic ellipse, let the area be put (AH). 
We must now express -<p and //, as functions of another variable 0. 
Let < 2 -= a cos 0, y =b sin0; the base of the cylinder being the ellipse whose equation 
2 2 
is \ J r ! j 2 == ^’ 'P * s the angle which */x 2 -\-y 2 makes with the axis a. 
Now 
and 
But 
therefore 
tand/=-=-tan0, 
~ x a ’ 
d4= 
abcffl 
g 2 cos 2 0 + b 2 sin 2 0‘ 
r 2 g 2 cos 2 0 + b 2 sin 2 0 
tan ^ 2— ^2 s 
(A 2 + a 2 ) cos 2 0 + ( k 2 + b 2 ) sin 2 0 
sec^=- 
k* 
Hence substituting these values in (437-)? we get for the area 
/ A T I \ ^ 
(AH )=3 
‘d8 [(A 2 + g 2 ) cos 2 0 + (A 2 + b~) sin 2 0] 2 A: 2 
[g 2 cos 2 0 + b 2 sin 2 0] 
■fl&f 
1G 
g 2 cos 2 0 + b 2 sin 2 0‘ 
Let 
G 2 — b 2 , G 2 — i 2 2 
G 2 +/t 2 ~ * 5 G 2 ~ e ’ 
* being the modulus and e 2 the parameter, as in (15.). 
The above expression may be written 
9 i /\ t t\ /A/0 n 
^ A _(g 2 cos 2 0 + 6 2 sin0) a/(A 2 + g 2 ) — (g 2 — 6 2 ) sin 2 0 
2g& A 2 g/0 
k a/ (A 2 + G 2 ) - (g 2 -A 2 ) sin 2 0 
ab 
Ir 
(g 2 cos 2 0 + V 2 sin 2 0) 
k \Z{k 2 + a 2 ) - (g 2 - b 2 ) sin 2 0 
b. 
d \ 
-It 1 
'(1^) 
(438.) 
(439.) 
(440.) 
(441.) 
(442.) 
( 443 .) 
