246 Mr. Ivory on the Properties of a Line of shortest 
tween the equator and the variable parallel; and we shall have 
these equations, sin A= sin? sina 
sin = sin z sin (a—s'). 
Now substitute this value of sin {, then 
ds=ds' ¥1 +e? sin*z sin® (a—s'); 
and this formula shows that the lengths of a geodetical line, 
reckoned from a fixt point on the surface of the spheroid, are 
equal to the arcs of an ellipse reckoned from a fixt point in 
the periphery. The greater semi-axis of the ellipse is equal 
to / 1 +e? sin? z, which is the semi-diameter of the spheroid 
perpendicular to the plane of the oblique great circle; the less 
semi-axis is equal to the radius of the inscribed sphere. 
Enough has now been said to show the use of the formulze 
in investigating the properties of a geodetical line. There can 
be no difficulty in this respect, at least if we suppose that the 
figure of the spheroid, or the proportion of the polar axis to 
equatorial diameter, is known. Without knowing this pro- 
portion, we cannot deduce the inclination of the oblique great 
circle to the equator, nor pass from the latitudes actually ob- 
served on the surface of the spheroid, to the corresponding 
latitudes on the surface of the sphere. In the question of the 
figure of the earth, the problem must therefore be viewed a 
little differently. It is necessary to introduce the angles ac- 
tually found by observation in the expression of the length of 
the geodetical line. Now, we have, 
cos 27 = COSA SIN»; 
and, by substituting the value of cos A, we get 
cos J sin & »/ T+ e2 
cos i = ——— — 
a/ 1 +e? cos?l 
A 1.— cos? / sin? # + e? cos? 1 cos? 
A 1 — e? cos21 
on a 
Let us now put 
cos 8 = cosZ/sinp: 
a/sin® B +e? cos? 1 cos? 
Jl +e cos2t é 
The are 6 being deduced from the latitude and azimuth on 
the spheroid, is always known; and it is very little different 
from 7. 
Again : if we combine the values of ds and d s', found in the 
formulze (A) and (B), we shall get, 
dy cos f / 1 +e2 sin? y) 4 
A sin? i — sin? Y 
then, sinz = 
ds= 
But, 
—— 
