to the Meridian at Beachy Head, $ 



the meridians of the two points, or the difference of longitude ; 

 then we shall have this equation 



■2L. = (a_ p cos pof+ p* sin 9 to -f- (q — t)* ; 



and, because cos co = 1 — 2 sin 9 -£-, the same equation may be 



thus written : 



£ = (p-uf + (q-tf + 4>pu sin 2 \. 



Now we have, 



p— w=(cosA— cos A') {i+ g— g(cos 9 A + cos 8 X'+cosAcosV)} 



q—t= (sin A— sinA') {1— 2e + e (sin 2 A + sin 2 A' + sin A sin A')}. 



And if we put m= -*-, n = ^~; ty substituting and 



neglecting the term multiplied by i sin 3 n, because n is a small 

 angle, we shall get, 



p — u = 2 sin n sin m ( 1 + s —3 e cos 2 7/2 cos 2 74) 

 2— * = 2sinw cosw(l — 2e -f3gsin*7» cos 2 w); 

 and hence, 



(p— u y+ (q—tf = 4?sin«» A + 2ssin 2 m — 4ecos 9 7tt)) i 



or, which is the same thing, 



(^-tt) a + (?-*) 2 = * sin 9 -%£ (l - 1 - 3g cos (A + A')). 

 Again we have, 



cos X COS X' 



pu = 



1— t (sin 3 X + sin* x') 



but as A and A' are nearly equal, we may take 2 sin 9 #£* = 

 1 — cos (A-f A') for sin 2 A + sin 9 A'; consequently, 



cos X COS X' 

 f U ~ l-t + icos(X-|-xy 



Lastly, we have a =jt?A ; A being the length of a degree on 

 the equator of the spheroid, and p the number of degrees in 

 an arc equal to the radius. All the values being substituted, 

 we get, 



y l • a*— x' /, „ /. , »i\\ cos X cos x' sin* g 



— — = sm 2 ( 1— e— 3gcos(A + A') )+ I 



4;*a» 1 V v '/~ 1-i-f icos(x-fx') ♦ 



From this expression the arc w may be computed by means 

 of these formulas ; viz. 



t . , /2nA . x — x' \ Mi 3Ms . n 



Log. sin u = log. f — - — sin — — j — 5- cos ( x + x " 



New Series. Vol. 4. No. 19. July 1828. C Log. 



