66 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
When a >b the integral is of the third order and circular form, but when a<b the 
integral is of the third order and logarithmic form. That it is of the logarithmic 
form may thus be shown. 
Let 
b 2 -a 2 , , b 2 
and *=oqT*- 
Hence 
1 m ~ b 2 {a 2 + b 2 ) ; 
or i 2 is greater than m ; but we know that the form is logarithmic when the square of 
the modulus is greater than the parameter, when it is affected with a negative sign. 
This is a result truly remarkable. All analysts know the impossibility of trans- 
forming the circular form into the logarithmic, or vice versa , by any other than an 
imaginary transformation. The utmost efforts of the most accomplished analysts have 
been exhausted in the attempt ; yet in this particular case their geometrical connection 
is very close. The modulus and the parameter are connected by the equation 
~+m= 2; (481.) 
the upper sign to be taken in the circular form, the lower in the logarithmic. 
There are two distinct cases to be considered ; when a is greater than h, and when 
a is less than h. 
Case I. a>b. 
Let a plane ellipse be constructed whose principal semiaxes A and B are given 
by the equations 
A 2 =a 2 +b 2 , B 2 =« 2 , (482.) 
and let a sphere be described from the centre of this ellipse with a radius 
a 2 B 2 
a — — — — R. 
V a 1 — b 2 V 2B 2 — A 2 
Then we can find, as follows, the length of an arc of the spherical ellipse, the intersec- 
tion of the sphere whose radius is R, with the cylinder standing on the ellipse whose 
semiaxes are A and B. 
Since 
and 
We have also 
. , A 2 « 4 -6 4 , i 4 
sin a=g 2 =— ^ 4 — 5 cos b=^ 4 > 
R 2 ffl 2_£2 £2 
si n 2 /3 = cos 2 /3=^> 
R cos/3 a 5 
cosa sina b(a 2 —b 2 ) ^a^^-b 2 
R cos/3 cosa 
sina 
ab 3 
(a 2 — b 2 ) x/^Tb 2 
„ cos 2 /3 — cos 2 a a 2 —b 2 
tana= — » = — jo— > 
cos^a b * 
sin 2 a — sin 2 /3 b 2 
? 2 =sin 2 p?= 
sin 2 a a 2 + b 2 
( 483 .) 
