Mr. Ivory on the Properties of a Line of shortest Distance. $41 
same thing by attending to what z represents on the surface 
of the sphere. Let the arc 7! be determined by this equation ; 
viz. . tan 7 
tan? = ieee. 
and draw a great circle having the inclination 7! to the equa- 
tor, and intersecting it in the same diameter with the former 
oblique circle. Now let any meridian meet the two circles, 
and let ) and uw be the arcs of the meridian between the equa~ 
tor, and the respective circles; then we shall have this equa-~ 
tion, v1z. tan w 
tany = Vis: 
whence it follows, that if / be the latitude of a parallel to the 
equator on the surface of the sphere, x will be the latitude of 
the same parallel on the surface of the spheroid. Hence it 
will readily appear that z is the are of the latter great circle 
intercepted between the two meridians that pass through the 
extremities of the arc s! of the former circle; and, on account 
of the proximity of the two circles, it is never much different 
from s' or s. When the meridian is nearly perpendicular to 
the circles, it is also evident, as has already been observed, 
that a small error in the latitude will occasion a great varia- 
tion in 2. 
Having expanded the foregoing formula and integrated as 
usual (using Hirsch’s tables of fluents, or the tables of any 
such plodding collector, if need be), we shall get, by neglecting 
the powers of e* above the square, 
2 sin? J ‘ : : 
> =m 1+ a — Gq (16 sin?2 — 13 sin* 7) ¢ 
: pin hg et ; ; 
+ sin 22 } SS sin’ _ 3S (4 sin*Z — 3sin*7) i 
15e sin4/ 
256 
and, in this formula, we have only to substitute the value of 
z in terms of the difference of longitude, or of the change in 
azimuth. ai : 
Let us first compare z with the difference of longitude. The 
second formula (A) gives us 
V1 +e2 sin? p ag! ¥, 
Wi Meee: ™ fl +e? costu 
Observing that here sing = 1, sinz = sin A, the formule (B) 
give us 
X sin 42: 
dg=q! x 
—d ¢! = 7 
cos Y o/sin®A—sin? y 
Now 
