DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Hence cosa tanX=tan(jW/+v). 
In (25.), or (38.) or (39.), we assumed tan'v^= cosa tanX. 
Hence ■4*=fjb-{-v, or tan('4/ — |U/) = tan{' = siny tan/>(-. 
A geometrical interpretation of Lagrange’s theorem tan('4' — iJA) = sin 7 tan^M/ may be 
given by the aid of the spherical parabola. 
Let DR^B be the great circle, the base of the 
hemisphere, whose pole is F. Let BQA be a sphe- 
rical parabola, touching the great circle at B, and 
having one of its foci at F the pole of the hemisphere 
whose base is the circle DR^B. Let RQ be an arc 
of a great circle, a tangent to the curve at Q. From 
F let fall upon it the perpendicular arc FR. The 
point R is in the great circle AR which touches the 
curve at its vertex A. The pole of this circle is the 
TT 
second focus F, ; for AF,=FB= 5 . Let the arcs 
RF, RF, make the angles (m and v with the transverse arc AB. Hence AR— y. In 
the spherical triangle FAR, right-angled at A, we have sinAF=tanj' cot/oo. Now as 
AF= 5 -, sinAF=sin 7 =:j ; and if ^ or reducing, tan ((p—|E//)=j taniO/ ; 
whence we infer that while the original amplitude is the angle (jij at the focus F, 
the derived amplitude (p is the sum of the angles and v at the foci F and F^. 
When the function is complete, or R will coincide with R, the pole of the 
TT 
great circle AB, whence v is also and as (p=r. This shows, that when 
Fig. 7. 
the function is complete, or the amplitude is a right angle, the amplitude of the 
derived function will be two right angles. 
When the spherical parabola approximates to a great circle of the sphere, the 
second focus F, will approach to F the immoveable focus. The arc RF^ will, there- 
fore, approach to coincidence with the arc RF, or the angle v will approximate to [Jj, 
so that (p=ijj-\-v=2(jj nearly. 
This is the geometrical explanation of the analytical fact observed in this theory, 
that when the modulus diminishes, or the spherical parabola approximates to a great 
circle of the sphere, the ratio of any two successive amplitudes approximates to that 
of two to one. 
When the transverse arc of the spherical parabola is a right angle and a half. 
sin 7 =-^, and if C be its circumference, C= V2 
^ -I 
'C~2 
-f-'T. But two qua- 
drants 2.s, or the loop of a lemniscate, are = V'2\' 
dju. 
Hence 2s=C—‘t. 
