332 Dll. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
or substituting the values of the constants given by the preceding equations, 
ff= 
\+j\ 
— sin®4' 
(61.) 
But when the focus is the pole, we found for the arc the following expression in (58.), 
J V 1 — sm> L V 1 — sin^ju, J 
Equating those values of a, we get the resulting equation, 
1+7 
r dvj/ 
U +jJ 
1 sin^v|/J 
|\/ 1-1 
a -A 
+jJ 
1 sin^i]/ 
j tanft 
XXIV. We shall now show that the amplitudes and in the preceding formula 
are connected by the equation 
tan (-v.^ — =j' tanjM;, (63.) 
a relation established by Lagrange. 
Let Ts and w' be the perpendicular arcs from the centre and focus of the spherical 
parabola on the tangent arc to the curve. Let \ and ^ be the angles which these per- 
pendicular arcs make with the major principal arc. The distance between the centre 
and focus of the spherical parabola, with the complements of those perpendiculars, 
constitute the sides of a spherical triangle. We shall therefore have 
sin^?i=sinV 
1 ! 
(64.) 
Now secV=sec^ce cos^>t+sec^i3 sin^X, as in (35.); or writing for seca, secjS their par- 
ticular values in the spherical parabola, given in (59.), 
2 
sec'w: 
Again, as 
' 1 — siny 
, tany 
tanw = 5 
COSjU. 
,_tanVfcosV . 
S 0 C t»y — — g 9 
■sin^X (65.) 
reducing (64.), the result is 
tan^>t=: 
2(1 -fsiny) 
(cotju, — siny tanju,)^' 
In the case of the spherical parabola. 
( 66 .) 
, 1 + siny 
cos^g= 
coss tanX= 
1 -)- siny 
2 , whence (66.) becomes 
tanfi. -f siny tanju. 
. . . . , or cosg tan>t=:T^^ — r- . / — i- /g^ \ 
cot/jt — smy tanju,’ 1 — siny tanfi, . tanju. ' ' 
The second member of this equation is manifestly the expression for the tangent 
of the sum of two arcs \ij and v, if we make tam'=siny tanjU-. 
