Distance traced on the Surface of an oblate Spheroid. 243 
Having now ascertained the import of the arcs ¢ and , all 
the properties of the geodetical line are readily deduced from 
the formule investigated in this Journal for July 1824. Using 
the arc { to denote the latitude (on the surface of the inscribed 
sphere) of a plane parallel to the equator which cuts the geo- 
detical line, put »! for the azimuth at the point of section; 
then the most distinguishing property is expressed by this 
- saan cos f! sin #’ = cosz (a) 
where cos z expresses a quantity which is constantly the same 
for every point of the geodetical line. If we suppose that the 
parallel plane moves towards the equator, and finally coin- 
cides with it, the foregoing equation will become, sin pu! = 
cos 7; whence we learn that the arc 7 is the inclination of the 
geodetical line to the equator where it crosses that circle. 
Conceive a great circle on the surface of the inscribed sphere, 
which is inclined to the equator in the same angle z; and it 
readily follows from the rules of spherical trigonometry, that 
the equation (a) is common to the geodetical line on the sur- 
face of the spheroid, and the oblique circle on the surface of 
the sphere; that is, the two lines have the same azimuth at 
any two points in the same parallel to the equator. It follows 
that a parallel which meets the oblique great circle, will like- 
wise meet the geodetical line; and consequently they are both 
contained within the same limits on either side of the equator. 
They both extend from the equator to two parallels of which 
the latitudes, on the inscribed sphere, are + 7; and having 
touched these planes, they bend back in an opposite direc- 
tion. 
In order to compare the two lines further, it is requisite to 
fix two initial points, or two points of departure from which 
to reckon. Having assumed any point in the geodetical line, 
draw through it a parallel to the equator. This parallel will 
likewise meet the oblique great circle; and we may suppose 
the plane of the circle turned about the centre of the sphere, 
till the point in the parallel comes to the meridian of the point 
assumed in the geodetical line. These two points, one in the 
oblique great circle, and one in the geodetical line, are the 
two initial points required; they are in the same parallel to 
the equator, and they have the same longitude. If a and Z 
denote the two latitudes of the parallel to the equator, the 
first on the sphere, and the other on the spheroid, we shall 
have, according to the equation before found, 
tan / 
Ji +e 
Suppose now that any other parallel to the equator cuts the 
2H 2 two 
tana = 
