Distance traced on the Surface of an oblate Spheroid. 349 
A=1 a 2 ee a a) 5f* + &e. - 
B=A-—1 
C=[B- Tf?  &e. 
The coeficients B, C, &c. decrease at the same rate with the 
successive powers of f*. It may be remarked here, once for all, 
that, if we are to calculate with the usual tables, the approxi- 
mation need not be carried further than to include /*: for it 
will be found that the other terms affect the numbers only in 
the eighth decimal place, which is beyond the reach of the or- 
dinary tables. This being observed, we have, 
(0) — ara pa = 15 
AMSA VI =fal+ 5 f+ Sf 
AM =BVI-f = if tis 
4 — sj by 
A® =C VI —ft = By 
s = Az — cosz fA) sing + A(3) sin?z}. 
Now this very simple expression will accomplish all that can 
be effected by M. Bessel’s formula (10) in p. 141 of the Jour- 
nal of Science. But it will be more convenient in practice if 
it be written a little differently, as follows, 
1 
m=— = 1— 
A 
Ae 3 
papa alt att 
eutth: (os allied 
tO) on aed 3 
ms = z—cosx fpsinz + gsin°z}. (2) 
For illustration 1 take M. Bessel’s example in pp. 143, 144, 
of the Journal of Science. The latitude of Seeberg, or J, is 
50° 56° 6""7; w, or the azimuth, 85° 38! 56"-82 reckoning 
from the north westward ; and, if s be the distance from See- 
a to Dunkirk on the geodetical line, and reduced toa sphe- 
roid of which the polar semi-axis is unit, we have, log s = 
8°9649485. Further, the square of the excentricity is, in my 
s 3 
notation, equal to ree and according to M. Bessel, 
e 
log ine = 7°8108710 
log e? = 7°8136900. 
From these data we get, by the formula (1), 
= 51° 4! g!94, 
And 
