THE 
PHILOSOPHICAL MAGAZINE 
AND JOURNAL. 
30 APRIL 1826. 
X XXVIII. On the Properties of a Line of shortest Distance 
traced on the Surface of an oblate Spheroid. By J. Ivory, 
Esq. M.A. F.R.S.* 
Y intention in treating of the geodetical problem inserted 
in this Journal for July 1824, p. 35, was to show that by 
giving a proper form to the coordinates of the surface of the 
spheroid, the usual analysis might be much shortened, and 
more simple formulz of solution obtained. If the polar semi- 
axis be unit, and / 1 + é represent the radius of the equator, 
the equation of the surface will be, 
+y? 
1+e? 
z being perpendicular, and x and y parallel, to the equator. 
Now this equation is satisfied by assuming 
xr=cos¢cos) /1 42 
y=singcos) V1 + 6 
z= sin, 
the angles ¢ and } remaining indeterminate. ‘The coordi- 
nates belong to a spherical surface when e = 0; and this man- 
ner of expressing them, which I have used on other occasions, 
is well adapted for simplifying the investigation of such pro- 
perties of the elliptical spheroid as are analogous to those of 
the sphere. 
With regard to the arcs and y, it is obvious that sin is 
the distance from the equator (estimated in parts of the polar 
semi-axis) of a parallel to the equator drawn through the point 
on the surface of the spheroid; and hence it is obvious that $ 
is the angular distance between the meridian passing through 
the same point and a given meridian. The arc Wis the lati- 
tude of a parallel to the equator on the surface of the sphere 
inscribed in the spheroid; but it is not the true latitude of the 
same parallel on the surface of the spheroid, as I have inadver- 
* Communicated by the Author. 
Vol, 67. No. 336. April 1826. des | tently 
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