DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 357 
making- wnth the plane of the circular section, or with the plane of xy, an angle whose 
. h 
tangent is the semiaxes A and B of this plane section will manifestly be 
e(2 — n) 
2^/1 —n 
If we denote the eccentricity of this plane ellipse by 
n 
h’ */ — 2 — k" 
B=a, and or A: 
— ? . 1 
==, or writings for V 1 
l-Z 
+i‘ 
HenceJ 
-1-*/ 
(180.) 
(181.) 
1+v'T^’ v - 1+*, 
It is shown in every treatise on elliptic integrals, (see Hymer’s Integral Calculus, 
p. 220,) that if c and are two moduli connected by the equation 
c,= 
Vl—c^ \ — b 
1+ V\—c^~'^ + b’ 
(182.) 
and (p and two angles related, as in (63.), so that 
tan(4 — ?>) = & tan^, (183.) 
we shall have (l + cJEc((p)=EcX'^) + ^;®^^'^~2 
2 46 
Now ]-}-c^=Y^, — hence 
E,(?)=^^^’E.,(+) + ^’sin4-A^F„W), (184.) 
and, using the common notation for the present, (74.) gives 
iFc(?>) = Y^F^/-4/). Adding these equations, we get 
E,(?) + 6F,(?) = ^E.,(4')+‘■V^sin^|-, (185.) 
or, using the notation adopted in this paper, 
( 186 .) 
since w = 1 — 5 = 1 —j. 
Substituting the value of the first member of this equation in (176.), the resulting 
equation will become 
(l+i) 
-Jd'J'yi, 
+? sin-^/ 
sirup cosp a/ I 
^cos^p -\-j sin^p ' 
Having put for its value in this case, namely, 
^ sinp cosp Vl 
” cos^p sin^p’ 
we must now combine the last two members of this equation. 
. . . . . 
2 cos-'p -{-j sin^p J 
(187.) 
Adding, they become 
( 188 .) 
