DR. BOOTH ON THE GEOJMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 377 
In the same manner we may show that 
JV l-e2 J 
sin^d] 2 
(303.) 
but 
Hence 


Substituting these values in (141.), we obtain 
''= [j.’ _ J’dfyi jJ ^+constant. (306.) 
To determine this constant. We must not suppose i=0, in this case, as is generally 
done, to determine the constant. This would be to violate the supposition on which 
we have all along proceeded, namely, that the variable cylinders are all similar, and 
therefore that i must be constant. We must determine the constant from other 
considerations. 
Since when a = 0, But e^=cos^0+*^ sin^0, therefore 6 = 2 - 
a, the major semiaxe of the base of the cylinder, is supposed to vanish, the curve 
diminishes to a point, and therefore <7=0. 
When a=A*, e^=l, and 0=0. We have in this case for the sections of a 
sphere by an elliptic cylinder, whose greater axis is equal to the diameter of the 
sphere, are two semicircles of a great circle of the sphere. Hence, when 0=0, 
0-=^, sin0=O, Jd0\/j=O, therefore the constant is equal to <7, when 0=0. 
But when 0=0, <7=^, or the constant is equal to 
The formula now becomes 
sin0 cos0“i 
\/J J‘ 
(307.) 
When 0=2’ ^^itl <7 = 0, as the variable cylinder is in this case diminished to a 
right line; therefore the preceding formula will become, using the ordinary notation 
of elliptic integrals, 
^=E,F,+E,F,-F,F, (308.) 
Hence we obtain the true geometrical meaning of this curious formula of verifica- 
tion discovered by Legendre. In its general form (307-), it represents the difference 
between the quadrants of a great circle and of a spherical ellipse. When the sphe- 
rical ellipse vanishes to a point, this expression must represent, as in (308.), the 
quadrant of a circle. 
LIV. If we now apply the preceding investigations to the curve described on 
3 c 2 
