DR. BOOTH ON THE GEOMETRICAL PROPERTIES OP ELLIPTIC INTEGRALS. 331 
y heing any arbitrary angle, may he rectified by an elliptic function of the first order. 
Write X for tana, y for tan|8, and eliminate siny from the preceding- equations, 
tan^a— tan^|8=x^— 1, (59*.) 
the equation of an equilateral hyperbola. We thus obtain the following- theorem; — 
Any spherical conic section, the tangents of whose principal semiarcs can he the ordi- 
nates of an equilateral hyperbola, whose transverse semi-axis is 1, may be rectified by 
an elliptic integral of the first order. 
XXII. When we take the complete function, and integ-rate between the limits 
0 and p, we get, not the length of a quadrant of the spherical parabola, as we do 
when we take the centre as origin, but the length of two quadrants or half the 
ellipse. We derive also this other remarkable result, that when is a right angle^ 
the spherical triangle whose sides are the radius vector, the perpendicular arc on the 
tangent, and the intercept of the tangent arc between the point of contact and the 
foot of the perpendicular, is a quadrantal equilateral triangle. For when jM'= 2 ’ 
It may also easily be shown, that the arc of a great circle which touches the spherical 
parabola, intercepted between the perpendicular arcs let fall upon it from the foci, is 
in every position constant, and equal to a quadrant. See Theory of Elliptic Integrals, 
p. 35. 
Hence the spherical parabola is the envelope of a quadrantal arc of a great circle, 
which always has its extremities on two fixed great circles of the sphere, the angle 
TT 
between the planes of these circles being 
Resuming the equations given in (59.), which express the tangents of the principal 
semiarcs of the spherical parabola in terms of siny, namely. 
tan^a = 
1 +siny 
1 — siny 
writing i for cosy, and J for siny, we get 
tan 2 = 
1-i 
2 '^—J-2 
er=zr-r-- sm5; 
=(lz/V 
(60.) 
1+/ - “1 VI +^7 ’ [^ 
whence tan^2=e^=sin;j=cos*/3. 1 
Now n=tan*£, m=e®; hence w=m= 2 .J 
XXIII. We shall now proceed to the rectification of an arc of the spherical para- 
bola, the eentre being the pole. By this method we shall obtain certain geometrical 
results which have hitherto appeared as mere analytical expressions. In (14.) or (28.) 
we found for an arc of a spherical ellipse measured from the major principal arc, the 
following expression, the centre being the pole, 
44/ 
taniS . .r 
sin^vj/) V I — sin^)j sin^vj/ 
