DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 339 
(97.) 
XXVIII. The foregoing investigations furnish us with the geometrical interpreta- 
tion of the transformations of Lagrange. Let the successive amplitudes p, % ot 
the derived functions, be connected by the equations 
tan(^-|M;)=jtan(M., tan(%|/— tan?), tan(%— tanx^/. 
We may imagine a series of confocal parabolas having 
a common axis, described on a plane in contact with 
a sphere at their common focus. These parabolas will 
generate a series of confocal spherical parabolas on the 
surface of the sphere, BCA, BC'A', BC"A", BC"'A"', 
which will all mutually touch at the vertex B remote 
from the common focus F. Let the distances between 
the common focus F and the vertices of the plane para- 
bolas subtend at the centre of the sphere, angles 7 , 7 ', 7 ", 
&c., whose cosines 
equations 
*//5 
&c. are connected by the 
1 - 
1 - Vl- 
i 2 
''III 
1 + V' I ■ 
. . &C., 
(98.) 
it is plain that 7 =FA, 7 '=FA', 7 "=FA", 7 "'=FA"', &c. 
We may repeat this construction successively, until the parameter of the last of the 
applied tangent plane parabolas shall become so indefinitely small, compared with 
the radius of the sphere, that it may ultimately be taken to coincide with its projec- 
tion. We shall in this way reduce, at least geometrically, the calculation of an 
elliptic integral of the first order to the rectification of an arc of a parabola, that is, to 
a logarithm, as in XX. If, on the contrary, the moduli i, &c. proceed in a de- 
scending series, the angles 7 , 7 ^ 7 ,^ continually increase, the magnitudes of the con- 
focal applied parabolas increase, till at length their parameters become so large, com- 
pared with the radius of the sphere, that their central projections pass into great 
circles of the sphere. The evaluation of the elliptic integral will therefore ultimately 
be reduced to the rectification of a circular arc. These are the well-known results 
of the modular transformation of Lagrange. 
The formulae established in (58.) for the rectification of the spherical parabola, give 
dfj. 
~ sin7j' 
Vl- 
-cos^y sin^ju. 
j-tan 
smy tanju. 
Vi- 
•cos^y sin/^_ 
or writing i for cosy,j for siny, and \/l for \/ 1 — /^sin^^, 
and t' being the corresponding quantities for the next derived spherical parabola, 
