DR. BOOTH ON THE GEOMETRICxlL PROPERTIES OF ELLIPTIC INTEGRALS, 375 
whose semiaxes are a and h. In this case the radius of the sphere is infinite. The 
least sphere is that whose radius is a, and which cuts the cylinder in its circular 
sections. Hence the greatest spherical elliptic quadrant is the quadrant of the circle 
whose radius is a. All the spherical elliptic quadrants will therefore be comprised 
between the quadrants of an ellipse, and of a circle whose radius is a. Any quadrant 
therefore of a given spherical ellipse is equal to a quadrant of a plane ellipse, 
plus a certain increment; or to a quadrant of a circle, minus a certain decrement. 
The same reasoning will hold as well when we take any other limits, besides 0 and 
These considerations naturally lead to the process of differentiation under the sign 
of integration, because we cannot express, under a finite known form, the arc of a 
spherical or logarithmic ellipse, and then differentiate the expression so found, with 
respect to a quantity which hitherto has been taken as a constant. 
We may conceive the generation of successive curves of this kind to take place in 
another manner. Let us imagine an invariable sphere to be cut by a succession of 
concentric cylinders indefinitely near to each other, and generated after a given law. 
These cylinders will cut the sphere in a series of spherical ellipses, any one of which 
will differ from the one immediately preceding, by an indefinitely small quantity. If 
we sum all these indefinitely small quantities, or in other words, integrate the ex- 
pression so found, we shall discover the finite difference between any two curves of 
the series separated by a finite interval. One of the limits being a known curve, the 
other may thus be determined. 
To apply this reasoning. 
In the following investigations we shall assume the generating sphere to be invari- 
able, and the modulus /, with the amplitude (p to be constant. The intersecting 
cylinder we shall suppose to vary from curve to curve on the surface of the sphere. 
fl2_52 
But i is constant, and ? see {27.). Now a and b being the semiaxes of the 
base of the cylinder, it follows that the bases of all the varying cylinders are con- 
centric and similar ellipses. Hence in the elliptic integral of the third order, which 
represents the spherical ellipse, the parameter ^ or m, and the criterion of sphericity 
,J K will vary. 
In (17.) we found for a quadrant of a spherical hyperconic section, the expression 
T 
Let k be the radius of the sphere. 
Since ^ vary, as I 
the variable cylinder. We have also 
e will vary, as being a function of a the transverse semiaxe of 
MDCCCLII. 
3 c 
