348 Mr. Ivory on the Properties of a Line of shortest 
cosi’ fl +e é 
V1+e cos?” 
wherefore, by equating the two values of cos 7, we get, 
cos? = 
4 ee cos / sin « 
COS = (Tp eee (1) 
Again, by combining the formulz (A) and (B), we get, 
ad cos f V1 +e sin? y 
c= ——— ; 
4/ sin*i—sin*y 
but, we have, 
a sin ? 4 A sin % 
smn z = SS, sin _ 
V1 + e cos *i : V1 + &cos*u” 
wherefore, by substitution, 
(1 + é) Vi+ecos*i. du cos u _ 
d § 5 s > . > 2 
(1 + e cos?u) 7 V sin 27 — sin 4x 
and hence, 
sin uw = sin 7’ sin z 
gees | +e)V¥1+e&cos*i.dx 
a+eé—eé sin ® i’ sin? x) 2 
This formula is different from that we have obtained above in 
no other respect,.except that s and z are now to be reckoned 
from the equator, the one along the geodetical line, and the 
other along the great circle of the inscribed sphere, having the 
inclination 2’ to the equator. The terminations of s and z are 
points in the two lines that have the same latitude w. We 
next obtain, 
; é& sin’ 7’ 
fi= Le? 
V1i—f?. dz 
=f ? sin ®x) : 
In order to integrate this formula, I assume, 
s 
Ww 
then having taken the fluxions and made the two values of 
ds coincide, I have found, 
oo — 
= Az—cosz{Bsinz + Csin*z + &c.} ; 
A—B=1 
: f PRRTES 
9B—3C= me 
— 3:5 44 
&c. 
and hence by exterminating B,C, &c. successively, we get, 
