116 Approximate Values of the Radii of Curvature, fyc. [Aug. 



Article VI. 



Approximate Values of the Radii of Curvature, #r . of the Elliptic 

 Arcs of an oblate Spheroid. By James Adams, Esq. 



(To Dr. Thomson.) 



SIR, Storehouse, near Plymouth, May 24, 1819. 



Should the contents of this paper merit a place in the Annals 

 of Philosophy, your inserting them therein will much oblige, 



Your humble servant, 



James Adams. 



1. PutC H = half the equatorial diameter, CK = half the 

 polar axis, — ^ — = e, or C H : C K :: 1 : 1 — C, 



s = sin. c = eos (radius unity) of latitude P, 



M = measure of a meridional degree in ditto 



P = ditto of a degree per. to meridian in ditto 



L = ditto of a degree of longitude in ditto 



m = ditto of a meridional degree at the equator 



_ C ditto of a degree perpendicular to the meridian ? _,-, . 



\ and of a degree of longitude at 5 



v — 57*29, &c. = degrees in a circular arc equal to radius. 



2. In the following conclusions, the terms affected with the 

 square and higher powers ofe are omitted. 



3. The pages referred to are those of the " Elements of the 

 Ellipse," &c. printed for Longman and Co. 1818, from whence 

 we have 



l + 3es* — 2e = radius of curvature corresponding to M,p. 81 



I + e a* = ditto, ditto P, 44 



c (1 + e s 9 ) = ditto, ditto L, 37 



1 — 2 e = (1 — ef = ditto, ditto m, 59 



1 = C H = ditto, ditto p, 45 



1 + e = -r— =s ditto at the poles. 



4. From the preceding article, we have 



- = — tzt.— = 1 +3es *> p = l+es>,- = c{\+es*), 



W = i + \ + eZ* e = * + 2ec % ~ P = d " 0', ande = 1 - 



s/ 



5. Therefore M - m = 3 em s 9 , m = t K = M(l -3es 8 ), 

 P - p = p e s°; P - M = 2 e M c 9 , L = pc (I + e s 9 ), p = 

 n ™> - = — , L = » c and 1 : c ;: P : L. From hence it 



(1 — e) a ' p p c' •» 



3 



