374 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
See Hymer’s Integral Calculus, p. 245. The parameters are reciprocal in this equa- 
tion, being 1 and 
Now this is the extreme case of the formula for the comparison of elliptic integrals 
of the third order and logarithmic form. We perceive that this formula results from 
the comparison of two expressions for the same arc of a common hyperbola. We 
may also see that it is the limiting case of the general formula for the comparison of 
elliptic integrals of the third order having reciprocal parameters ; a formula which in 
like manner has been deduced from the comparison of two expressions for the same 
arc of the logarithmic hyperbola. It is also evident that being the difference 
V 1 
between tan(p\/l and 
simp cosp 
it is the difference between tangents, one drawn to 
the hyperbola, the other to the plane ellipse, for tamp^/l denotes the portion of a 
tangent to a hyperbola between the point of contact and the perpendicular on it 
from the centre; and ^ denotes a similar quantity in an ellipse; this differ- 
ence is precisely analogous to the expression which occurs in (284.) 
which denotes the difference between two parabolic arcs, one drawn a tangent to the 
logarithmic hyperbola, the other a tangent to the logarithmic ellipse. 
Section VII . — On the Values of complete Elliptic Integrals of the third order. 
LIII. We have hitherto investigated the properties and lengths of elliptic curves, on 
the supposition that the generating surface, whether sphere or paraboloid, was inva- 
riable, and that the lengths of the curves were made up by the summation of the 
elements produced by the successive values given to the amplitude <p between certain 
limits, 0 and g, suppose, if the arcs are to be quadrants. Thus the length of the 
quadrant is obtained, by adding together the successive increments which result from 
the successive additions, indefinitely small, which are made to the amplitude. We 
may, however, proceed on another principle to effect the rectification of those curves. 
If, to fix our ideas, we want to determine the length of a quadrant of the spherical 
ellipse, we may imagine the vertical cylinder, which we shall suppose invariable, to 
be successively intersected by a series of all possible concentric spheres. Every 
quadrant will differ in length from the one immediately preceding it in the series, 
by an infinitesimal quantity ; and if we take the least of these quadrants, and add to 
it the successive elements, by which one quadrant differs from the next immediately 
preceding, we shall thus obtain the length of a quadrant of the required spherical 
ellipse, equal to the least quadrant which can be described on the elliptic cylinder, 
plus the sum of all the elements between the least quadrant and the required one. 
Thus, for example, the least quadrant which can be drawn on an elliptic vertical 
cylinder, is its section by an horizontal plane, or a quadrant of the plane ellipse. 
