1819.] of the Elliptic Arcs of an oblate Spheroid. 117 



appears, that both M — m and P — p vary in a duplicate ratio 

 of the sine of latitude nearly, 3 e m and p e being given. 



6. At the poles s = 1 and c = ; then from the above expres- 

 sions, we deduce M — m = 3 e m, P — p = p e, P = M, m = p 



(rir e ) = p C 1 ~ 2 e ^>P = m (ttt) "C 1 + 2e > and L 



= 0. 



7. An approximate value of e may be found by the following 

 proportion ; viz. 1 + 3 c s 9 — 2 e : 1 + 3 e S s — 2 e :: 1 -f- 

 3 e s* : 1 + 3 e S 2 :: M : M' (Art. 1 and 3) from whence e ~ 



3 (M s*~- M' »') near ty 5 where M' and M represent the lengths of 



a meridional degree in two given latitudes, the sines of which 

 being S and s respectively. 



8. Or an approximate value of e may be found as follows ; by 

 Art. 3 we have 1 — 2 e + 3 e s 2 : 1 + es 9 :: M : P, or by substi- 

 tuting 1 — c 9 for s a , it will be 1 — 3 e c 2 + e : 1 — e c a + e :: 

 M : P, or 1 - 3 e c 4 : 1 — e c 2 : M : P, from whence e = 



P — M 



p _ M t| t nearly, M and P being the lengths of a meri- 

 dional and perpendicular degree in any given latitude. 



By omitting e in the above proportion, the denominator is 

 evidently too great, but may be reduced by writing 3 M for 3 P 

 (P being greater than M, p. 150 " Elements of the Ellipse ") we 



P — M 



shall then have e — — —, — , nearer than before, but not so 



2 jvi cos. a tat. 



correct as given in Art. 7. 



9. By having the measure of a meridional degree at the equa- 

 tor and the compression, to find the equatorial diameter and 

 polar axis. Put C H for half the equatorial diameter, and C K 



for half the polar axis, then (Art. 3) — — = — — x C K = — ^-' 



c ii c h i 



lCK=i..'.CK= ji^ • but by Art. 1, 1 - e : 1 :: C K 



in C H' C H /~< tt 1 «'»' imp 1 



10- -7T17- = rrrr X C H = ; X „ = X 



CK CK I ,- e (i — *>* m 1 - e ~~ 



-^—— (Art. 4 and 9) equal to the radius of curvature at the poles 

 (Art. 3). Now if M represent the measure of a degree at the 

 poles, then we shall have -^- = v M ther. M = -r— , and e 



* 1 — e 1 — e' 



ss \ — ~. By comparing the value of e in Art. 4 with the 



above, we have y/J m £ .-. M = p y/£ 



11. The quudrantal arc of the elliptic meridian is equal to 



