244 Mr. Ivory on the Properties of a Line of shortest 
two lines: let be the latitude of the parallel on the sphere; 
s' the arc of the oblique great circle between the parallel and 
the initial point; and 4! the spherical angle subtended by s! 
at the pole of the sphere, or the difference of longitude be- 
tween the extremities of s'; also let ~ be the azimuth of the 
oblique circle at the initial point, and yp! the azimuth at the 
other extremity of s': then we shall have the following equa- 
tions, VIZ. cos 2 = cosA sin 
ists dw cos 
A sin?i — sin? wy (B) 
do! — St) EEN SO 
cos y s/sin®?i — sin? y 
dp! aS cos id sin > get tl 
cos Y /sin?i — sin? ; 
These formulee express the relations between the differentials 
of the latitude, longitude, and azimuth of a variable point in 
a great circle of the sphere having the inclination z to the 
equator; and their use is to compare them with the like quan- 
tities in the geodetical line. The expressions of ds! and dq! 
are the same with those marked (B) in the communication in- 
serted in this Journal for July 1824, except that I have here 
written sin?z— sin*) for the equivalent quantity cos?) — 
cos*z = cos*  — cos*Asin® yw. The expression of d p!, which 
is common both to the circle and the geodetical line, has now 
been added. 
Again, let s denote the part of the geodetical line cut off by 
the same paralle] to the equator; and put ¢ for the difference 
of longitude of the two extremities of s; that is, for the angle 
containéd between the meridian which passes through the va- 
riable extremity of s and the fixed meridian of the two initial 
points. Then, according to the formulz marked (A) in the 
communication alluded to, we shall have, 
ds=d3!x f1+esin*y, 
ea A) 
rete f Vi+e%siney | ( 
dg=d}' x pager 
These different formule, extremely simple, contain all that is 
necessary to a complete theory of a geodetical line on the sur- 
face of an oblate spheroid. Ihave here merely supplied the 
geometrical explanation of the analytical solution before given. 
Let us compare the longitude in the geodetical line with 
that in the oblique great circle. The second formula (A) 
shows that d $ is always less than d ¢!; and hence the longi- 
tude in the geodetical line continually falls behind the longi- 
tude in the great circle, the defect accumulating more the 
further 
