260 



Archdeacon Pratt on the effect of 



dent variables, but are functions of u and v, and of the deflections produced 

 by local attraction. In the preceding diagram the plane of the paper is 

 the plane of the meridian in which the arc, of which A B is one section, 

 has been geodetically measured. A is the reference-station of the several 

 portions of the whole arc. A Z is the vertical at A in which the plumb- 

 line hangs. The two curves, of which A'B' and ah are portions, are a 

 variable eUipse and the mean ellipse having the same centre O and their 

 axes in the same lines, the mean ellipse being what the variable ellipse 

 becomes when the values are substituted for u and v which make the sum 

 of the squares of the errors a minimum : Z'A AN' and A aN are nor- 

 mals through A to these two elhpses ; AD, A'w', am are perpendicular 

 to OD. 



Now, if the earth had its mean form, a plumb-Hne at A would hang in 

 the normal zK to the mean ellipse ; but it hangs actually in ZA. Hence 

 ZA^r is the deflection (northward in the diagram) which the plumb-line 

 suff^ers from the local attraction arising from the derangement of the figure 

 and mass of the earth from the mean. This angle is some constant but 

 unknown quantity t, t being reckoned positive when the deflection is north- 

 ward. This quantity t is part of the correction ZAZ', or added to the 

 observed latitude of A before applying the principle of least squares. The 

 other part is -s-AZ', which I will now calculate : it is the angle between 

 the two normals drawn through A to the variable and the mean ellipses. 

 By the property of an ellipse of which the elHpticity is small, 



0N=2e.0m, and 0N' = 2e'.0m'. 



Also as Om, Om', OD differ only by quantities of the order of the ellip- 

 ticities, they may be put equal to each other in small terms, because we 

 neglect the square of the ellipticities. 



.-. ZMZ'=ZNAN'=ZAN'D-ZAND 



_^ cot AND-cotAN'D __ (ND— N'D)AD 

 1+cotAND'cotANb AD^ + ND.N'D 



. , (ON -ON)AD 7 2(e-e)0D.AD , . 



^^^'^ AD^ + DO- ad4dO- ~"=^^^" (^ -")^^^^^ 



sin 1 



= (e'— e) sin 2/-^^^, I being the observed latitude of A. 



Suppose that v and V are the values of v for the variable and the mean 

 ellipses. Then by the third of the formulae (1), 



r5oiSF(^-^)=13'''75sin 



Hence 



a!=t-\-n(v—'Y), 



