Neill. — On the Deflection of the Plumb-line. 583 



T 



Similarly, if p = 



4 1 

 8' = p sin - — - p l sin ^ + &c (15) 



To take a numerical example, suppose that in latitude 45° S. it is 

 required to find the difference in latitude between the top of a hill, 

 490 metres high, and the sea-level vertically below it, using Helmert's 

 spheroid. 



The value of 6 can be computed by equation (3), but it can be found 

 directly from (12), (14), and (15), as under. 



t = 8 + S = 8'+S' 



S' -S = 8-S' (16) 



Now, S' c = ^ + 8. 



.-.8' O -S O = 20. 



or = \ (S' - 8J (17) 



And by the aid of (14) and (15) 6 in seconds of arc is 



(^ Ift-Lfdn-l-l ^t-llsin^-f&c (18) 



2 sin 1 2 4 sin 1 v y 



yi -p = ^-J = r ^ ~ ^ = 10 - 6 x 893. 



Computing the first term of (18) from the known value of sin ^ we find 



6 = 0"-053 

 Thus the latitude of the hilltop is 



= 44° 59' 59"-947. 



These differences of latitude are so small that in most cases they will 

 be masked by the greater differences due to local conditions, which deflect 

 the plumb-line ; they possess the advantage that they can be easily com- 

 puted, and the observed value can be corrected or reduced to the sea-level. 



A summary of the three principal results is as follows : — 



(1.) A plummet suspended from a point at the sea-level deviates from 

 the vertical in the plane of the meridian, except at the Poles or the Equator ; 

 the amount of the deviation depends on the length of the suspending cord, 

 and the deviation it towards the Equator. 



(2.) A plummet suspended from a point above the surface of the sea- 

 level, when the suspending cord just reaches the surface, always coincides 

 with the vertical for any length of the suspending cord. 



(3.) At heights above the surface the observed latitude is less than its 

 corresponding value at sea-level. 



