i 

 \ 



Bowditch on the Oblateness of the Earth, 43 



V 



■ 



g-. sin. S. (ang. tang. \) +— . sin. 2 (ang. fan.-) — ang. (an. a=0, 



from which we obtain, by a few operations with Sherwin's or Hut- 

 ton's logarithms, the value S=t2;5S93. 



A 



SECTION THIRD. 



In Book 3, § 38, of the Mecanique Celeste the form of an os- 

 cillatory ellipsoid, corresponding to any part of the earth's surface, 



r 



is investigated, supposing the radius r drawn from the centre of the 

 ellipsoid to any point of its surface to be represented by 



r=l— c,sin."f^{l+A.cos. S(?>+e)} (1) 

 in which %|/ is the latitude of the place, 9 its longitude counted 

 from a fixed meridian, a, hf C constant quantities depending on tlie 

 form of the earthy a being of the same order as the elliptfcity of 

 the earth. At the equator of this ellipsoid where 4' = 0, r be- 

 comes equal to unity, corresponding to an ellipsoid of vevolutiony 

 f and at the pule where >(. = 90% it becomes 1 — a — a 7^ cos 3 



{9 -f-^jy which is not constant, as it ought to be ; since it contains 

 the variable quantity f. Therefore both these extreme values of 

 r are defective ; the one because the ellipsoid is limited to the case 

 of j^evolution^ the other because the polar axis is variable. To 

 correct this we must add the term a h. cos, 2 (9 4 C) to the gener- 

 al expression of r, as we shall now proceed to show. 



The equation of the earth's surface, assumed hy La Place in 

 his Mec. CeL Vol. 2. Pag. 109, is u == 0, which, in page US, is 

 reduced to the form ^ x^ -^y^ + z^ — 1 — t ait', a?, ^, z^ being 

 the rectangular coordinates of that point of the earth^s surface 

 above mentioned, whose distance from the centre is r, so that 

 a?^ +y^ -hz^^r^. Substituting this in the preceding equation we ^(^i 

 r^ = 1 +2 a a% and by neglecting the secoud and higher powersof r, 



7 



r = i+ « u'. (3) 



♦ 



