ON THE MEEIDIAN. 'Ill 



« 



ni.tcly fmall part of the ellipfej then if A F = z, GF=zzt.hQ fluxioa 



of £he arc i F, And o! G H be drawn, then the an.o-Ie G H F = A ih^ 



fliixion of the arc of latitude to rad. i. — Hence 215 i : A '. :FHVz:^ 



A 4- F H. 'B>u.t the radius of curvature F H = a" 6^ {a^ — £5* . Sin* 



A 4- ^* . Sin.* A) Then i^ c z=za — 3 we have b == a —c, zrd d^- =z 

 a^ — 2 a c -{. c' =:&* — 2 a c nearly f nee c is very fmall corrsparcd 



r 

 '■- ' ■■*"" "^ 



with a or 5. — Hence F H = a^ (a — a c) , (a^ — 2 a c . S'lti"' A)} 

 But (a* —^ 2 a c . Sin.* A) expanded is equal to ^ (i + . Sin.'' 



./^) nearly, by rejecting all the terms involving c" and therefore i^"// 

 =: a — 2 c +3^. Sin.'' ^, which fubllituted for F H, v/e get z =. A 

 [a — 2 r + 3 c . Sin.*yl) =:z A [a —- 2 c) + A { S ^ • Sin.' A.) But 



1 — Cos, 2 A , , ^ 



Sin.* A = — —.—._-. and therefore z == A (a, — 2 c) -{'' ^ . c A — 



I- . c A . Cos. 2-^ ? whofe fluent is 2 ==la — /O.^--— " i ^ • Sin^ 2 ^ .== 

 a A — c (^ 4- I A\ Sin. 2 ^) which requires no corredion ; and this is 

 the meafure of an arc ori' the meridian exteradin^g- from the-ecju^tor to 

 the latitude of the point F, where A denotes the arc of latitude in parts 

 of the rad. i, 



Lbt N be any other point whofc arc of latitude is A, Then A N = 

 a A-^c j— 4— A . Sin. 2 A^ and hence we get TN = fl {A -— A) -^ c 



\ 



+ I Sin. 2 A ^ % Sin. 2 a\ Put A -^ A = w, — -Jl 4. f Sin; 



2 ^ -- -] . Sin, ii ^ = ;7,and L the length of the meafuredarcin fathoms; 



Gg 



