396 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
On Conjugate Arcs of a Spherical Parabola. 
LXVII. The well-known relations between elliptic integrals of the first order, 
whose amplitudes are conjugate, develope some very elegant geometrical theorems. 
Thus in fig. (25.), since the arc AQ=jfi-l-QR, and the arc 
the arcs AQ+BQ=j[j’^+|^] +QR+QR' (a.) 
Now AQ-|-BQ= two quadrants of the spherical 
parabola, and QR-f QR'=^, whence half the cir- 
cumference, or AQB=j[J^+J^]+^- 
In XXII. it has been shown that the complete 
integral represents the semicircumference, whence jji 
''2 d«J . TT 
Fig. 25. 
AQB=jf' 
Jo 
(b.) 
Comparing these equations (a.) and (b.) together, we get 
fa do; r dx 
Jo 
Now as the triangle RR'P is a quadrantal right-angled triangle, the relation 
between the angles AFR, BER', or (p and %, is easily discovered. Since FPE is a 
spherical triangle right-angled at P, 
and FE= 2 £= 2 — y, we getj tamp tan%: 
When AQ=BQ, <p=x, and tan<p=^|^- 
The locus of the point P is a spherical ellipse, supplemental to the former, having 
the extremities of its principal minor arc, in the foci F, E of the former. 
LXVIII. Let (T, be three arcs of a spherical parabola, corresponding to the con- 
jugate amplitudes <p, %, &>. Then successively substituting these amplitudes in (58.), 
the resulting equation becomes 
But as the amplitudes ?>, %, a are conjugate, the sum of these integrals of the first 
order is 0, whence 
(353.) 
Or, when the amplitudes of three arcs in the spherical parabola are conjugate ampli- 
tudes, the sum of the arcs is equal to the sum of the protangents. We use the word 
sum in its algebraic sense. 
