DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 397 
On Conjugate Arcs of a Spherical Ellipse. 
LXIX. If, in (42.), we substitute successively <p, x-, and add the resulting equa- 
tions, we shall have 
t-* r 
rdp 
r 
mn L, 
Now the conjugate relation between p, % and m renders the sum of the integrals of 
the first order =0, and the sum of the integrals of the third order equal to a circular 
arc 0, which is given by the equation 
tan0= (365.) 
1 — ^ COSf COSX COSco 
Hence ff+ff;— ff,(=0 — r— (356.) 
Or, when the amplitudes are conjugate, the sum of three arcs of a sphei'ical ellipse may 
be expressed as the sum of four circular arcs. 
When one of the amplitudes a; is a right angle, becomes a quadrant of the sphe- 
rical ellipse=ff. 7-()=0, and 0=r=r^, as we shall show presently, whence 
(ff— (7,) — (7=r, which agrees with (52,). 
Or the difference between two arcs of a spherical ellipse, measured from the vertices 
of the curve, may be expressed by a circular arc. In (45.) we found 
mn simp cosp mn cos;^ 
tanr=:- 
tanr,=- 
COSip 
1 — P sin^p ’ ' 1 — sin^‘ 
% 
cosx 
Now when ^= 5 , (338.) gives sin>;= sinp= 
whence sin^ sinv= ^ 
or 0=T=r^, when r^, = 0, or 
LXX. When we take the negative parameter m instead of the positive w, ( 17 .) gives 
Now the sum of these arcs is equal to a circular arc— 0^, which may be determined 
by the expression 
tan0,= ,g 5 g . 
* rn ^ ' 
1 +: 
^ - cosp cos;^^ cosw 
whence cr-l-o-^— <7^^= - 0^ ' (359.) 
If we compare together (356.) and (359.), we shall have the following simple rela- 
