Distance traced on the Surface of an oblate Spherotd. 247 
But, u and ) being the latitudes of the same parallel to the 
equator on the spheroid and the sphere, we have, 
. v ia sin u me 
Ge ie ea ap 
and if we transform the expression of ds, by introducing the 
values of sin { and sin 2, we shall obtain, 
A= V sin? 6 + e? cos? cos? » — sin? u (1 +e? cos*/ cos’ p)|, 
He 2 (1+e2) f/f 1+e2 cos22.ducosu ; (C) 
(1 + e2cos?u)?. A 
This formula determines the length of s by means of the lati- 
tudes at the two extremities, and the initial direction with re- 
spect to the meridian. 
In the first place, if the geodetical line be in the direction 
of the meridian, then sin » = 0, cos » = 1, sinB = 1: hence 
A =cosu 7 1 + e*? cos? J, 
1+e2)d 
ds — gk 2) oe . 
(l+e? cos? uw)? 
If we expand the radical and integrate as usual between the 
limits 7 and u, supposing w greater than /, we shall get, 
— — (sin 2% — sin 21) 
— ets 3(u—) __ 3(sin 2u — sin 2) 15(sin du — sin 42) 
u—l 
= =u—l+e?} 
64 32 . 256 
I have here written > for s, on the supposition that s is the 
actual length measured, and P the semi-polar axis expressed 
in the same parts. There are therefore two unknown quan- 
tities P and e*; and consequently two different measurements 
are required at different latitudes. The latitudes chosen ought 
to be very distant from one another, one near the equator and 
one near the pole, in order that the curvatures of the meridian 
may be as different as possible. In places not remote, the pro- 
portion of the lengths measured would approach so near the 
proportion of the observed differences of latitude, that the un- 
avoidable errors of observation would render the result quite 
uncertain. 
Let us next suppose that the geodetical line is perpendicu- 
lar to the meridian. In this case, sin » = 1, cos » = 0, sin 
6 = sin 7, and A= 9 sin*/ — sin* u 
Wherefore, if we put sin « = sin Z cos z, then 
A= sin 7/ sin z, 
d gis (1+ e?) f1l+e* cos?l. dz 
(1 +e% — e2 sin ?1.cos? x)? 
In 
