Distance traced on the Surface of an oblate Spheroid. $43 
F) sin? 
perfi+¢sa- - + sin*/sin* x) 
= ot (oes, NE py SBE 
8 
2 64 
. 2 . 4 . . 
+sin2e {xs sin?7 — (4. sin®2— 3 sin * 7) t 
15 e! sin J 5 
256 xX sin 42. 
In this formula the only unknown quantities are e” and P, all the 
coefficients being known, provided the initial latitude and the 
difference of longitude have been determined by observation. 
If the length measured extend to an amplitude of only 2° or 
°, we may make cos z = 1 in the denominator of the diffe- 
rential equation, and then, 
In this case we likewise obtain from the formula (D'), 
z=uw /1-+ é cos*l; 
and hence, ae Oe Se 
V1 pe cos*t 
The quantity into which wz is here multiplied is the radius of 
curvature of the geodetical line; or, it is the normal to the 
surface of the spheroid terminating in the axis of revolution. 
Let us next compare z with the change of azimuth. From 
the fundamental equation (a) we get, 
cos A cosl ,a/1l+e?cos?u 
Put w = 90° — yz’; then, 
cos u 1+ e- cos ah 
sin p! = 
sin w = 
Consequently, 
a Masher sin / sins i: 1 “A 
AV cos*1 + sin?/ sin*® z a/ 1+e* cos? 
tan w a/ 1 +e2 cos2t 
tant / T—e?cos?/tan?w 
It appears therefore that z is rigorously determined by the 
change in azimuth. But it will be better to have recourse to 
approximation. Assume 
sin z = 
, . AK Sa Les 
z= sin a ————————— 
then sin K/] Jt —esin?/sin?y ‘ 
or sine = sin y / 1 + & cos? ll +  sin®Z sin*y. 
And, 
