582 Transactions. 



be 

 Equation (9) represents a circle of radius j— — > the co-ordinates of the 



, . be 



centre being y = o. x = =— 



w 3 b + c. 



If in fig. 1 the inner ellipse represents the surface of the earth and A 

 a point at an elevation above the surface, from which a plummet is suspended 

 by a cord Ap, the line Ap produced bisects the angle FjoF 1; and is normal 

 to the surface of the earth at p ; and the point A is therefore vertically 

 above p. This condition holds for any length of the suspending cord, 

 and the plumb-line at the earth's surface suspended from a point above 

 it always coincides with the vertical. 



The level surface at p is at right angles to Ap, and the angle EH?) is 

 the geographic latitude. 



The level surface at A is at right angles to the line AK, which bisects 

 the angle FAF 15 and the geographic latitude of A is the angle EKA. The 

 difference of these angles is the variation of the latitude of p, due to the 

 spheroidal form of the earth, and the elevation pA. 



The level surfaces become more spherical as we recede from the earth, 

 the point K moves along the major axis and ultimately coincides with the 

 centre. 



If </>j denotes the latitude of p and <f> 2 the angle EKA, then <f> 1 - <f>» is 

 the angle 6 in equation (3), where A is the origin, and the line bisecting 

 FAFi the initial, or the axis of X. 



Let3 = HAF. S = AFp. S' = HAFi. S' = AF,p. 



FpF x = »//. AP = r. 

 ¥p=U F.P^/ 



It is now proposed to express AF, AF,, and 8 , S' Q in terms of pF, pF h 

 and r. 



The triangle AFp gives 



AF 2 =f t 2 + r 2 + 2fs cos % (10) 



j x sin S = r sin S (11) 



S + S = f (12) 



Similar expressions are obtained from the triangle AF,pfor AF,, S' 0J and 8'. 

 From (11) 



T 



sin 8 = ~ sin o 

 h 



p 1 sin 



(|- s), where Pl = j (13) 



For even the highest mountain-tops the values of p x and 8 are small, and 

 equation (13) can be developed by Maclaurin's series, thus : 



■A 



1/ Pisin^ \ 

 V 1 + Vi c « s J 



8 = tan 



\ 1 4- 11. P.nS 



2- 



. if/ 1 

 = Pi sin 2 ~ 2 Pl ' sm ^ + &c ^ M) 



