Relating to the FIGURE of the EARTH. *j 



hs treated as fmall quantities, or mere fluxions of the earth's 

 circumference, the calculation muft be made by rules qviite dif- 

 ferent from thofe that have been hitherto employed. Thefe 

 new rules are deduced from the following analyhs. 



4. Let the ellipfis ADBE (fig. i. PI. I.) reprefent a meridian 

 pafling through the poles D and E, and cutting the equator in 

 A and B. Let C be the centre of the earth, AC, the radius of 

 the equator, rr a, and DC, half the polar axis, =: b. Let FG be 

 any very fmall arch of the meridian, having its centre of cur- 

 vatui-e in H ; join HF, HG cvitting AC in K and L. Let p 

 be the meafiire of the latitude of F, or the meafure of the angle 

 AKF, expreffed, not in degrees and minutes, but in decimals 

 of the radius i ; then the excefs of the angle ALG above 

 AKF, that is, the angle LHK or GHF will be = <p, and there- 

 fore YG =z px FG. Alfo, if the elliptic arch AF = 2, FG = 

 z = ipx FH. 



But FH, or the radius of curvature at F, is =: 



■=z a-" h'' [O' — «'fin'(p + 3*fin'<p)~^, as is 



(a' — a' fin > + 6' fin V)"" 



demonftrated in the conic fed:ions. Therefore, if c be tlie 

 compreflion at the poles, or the excefs of a above b^ 

 b'^zza'^ — 2rtc + c% or becaufe c is fmall in comparifon of a, if 

 we rejecfl its powers higher than the fir ft, 3' =: «* — lac^ and 



FH — a^{a — 2c) (a* — a"- fin '<p-f- «* fin '^ — lac fin Y) ""^ = 

 «' [a — 2c) [a^ — 2flcfin '(p)~^. 



But («' — 2rt<rfin'(p)-^==<? — 3(i_^fin^^)— ^ = 

 rt — 3 (i +^fin'<p) nearly, rejeding, as before, the terms that 



involve c\ &c. Hence FH = (a _ 2<:) (i + ^fin ^p) = 



rt — 2c + 3C- fin ^(f). 



Now, 



