RELATING TO THE FIGURE Ct THE EARTH. 105 



its centre of curvatulre in H ; join HF, HG cutting AC in Inveftlgation of 



K and L. Let (p be the meafure of the latitude of F, or the ^"'"""'f.' ^' 

 ^ ' computing the 



meafure of the angle AKF, exprefled, not in degrees and figure of the 

 minutes, but in decimals of the radius 1 ; then the excefs of "'^*' 

 the angle ALG above AKF, that is, the angle LHK or GHF 

 will be = <p, and therefore FG = <P X FG. Alfo, if the 

 elliptic arch AF = t, FG = t = ?* X FH. 



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



3 = «* b"- (a* - a* fin *(p + 6* 



(a»^ a* tin ''<?) + 6* fm *?>)» 



fin*?!) *, as is demonfirated in the conic feftions. There- 

 fore, if c be the compreffion at the poles, or the excefs of a 

 above b, b^ =za* — 2ac + c*, or becaufe c is fmall in com- 

 parifon of o, if we rejed its powers higher than the firft, i* == 

 a* — 2«c,and FH = a» {a — 2c) (a^ — a^ fin »(?> -f a' fin* 



(p — 2ac fin 2?>) ^ = a^ (a - 2c) (a* - 2ac fia »?») ** 



3 q Qc 3 



But (a^ - 2ac fin «^) "Z" = a "^ (1 fin «?)) * 



a 



= a ■ (1 -}- — fin *^) nearly, rejeding, as before, the 



3c 

 terms that involve c*, &c. Hence FH = (a — 2c) (1 -| 



fin •?>) = a - 2c -f 3c fin 2^. 



Now 



