344 Mr. Ivory on the Figure ofthe'Earih, as deduced from 



If we now put */ 1 *- e % « 1 — •«, then e will be the compression 

 or the ellipticity; arid we shall have, 



a(l-t)*.dk 

 (l_(2 s _ 8 2) sin*\)4 



By expanding this formula and integrating as usual, rejecting 

 the cube and higher powers of e, we shall get, 



f = «• l^- £ (i + T sin2A ) + sS (^ +-M- sin * A )}' 



And, if p denote an arc of the meridian between the latitudes 

 V and X ; then 



<p = a . { A- A'-£ (j±^ + -| (sin 2 A - sin 2 V)) 



Further, let n = A — A' 

 m = x + A' 

 ^ = 57°-2957795 



A = f \ 



then, $ = A . | w — s (~ + ~ p sin n cos w) (A) 



+ 8 \-Tq + Tg" i 7 sm n cos w cos 2w ) \ * 



In this formula A is the length of 1° on the earth's equator; 

 and n and the coefficients of g and e 2 , are reckoned in degrees. 

 There has been so much discussion about the merits of the 

 different portions of the meridian that have been determined 

 trigonometrically, and their character is so well known, that 

 it would be superfluous to say any thing on that subject here. 

 The following table, which is taken from a paper by Professor 

 Airy, in the Phil. Trans, for 1 826, contains the five arcs which 

 unquestionably deserve the preference, both as they are the 

 longest that have been measured, and likewise on account of the 

 excellent instruments employed, and the great care taken in 

 executing all the operations. But even of these there is one, 

 namely the Swedish arc, measured by Svuanberg, to which 

 some objections are made ; because it makes the length of a 

 degree in latitude 66° 20', more than 200 toises less than it 

 had been found to be by Maupertuis and the French Acade- 

 micians who accompanied him. I shall therefore leave out 

 this arc in determining the elements of the elliptical figure of 

 the earth ; but, these elements being found by means of the 

 other four arcs, we may then apply them to the case omitted, 



both 



