18 Archdeacon Pratt on the Determination of the 



are applied to the observed latitudes, so as to make the lengths 

 of the corresponding divisions of the arc fit an ellipse; the same 

 is done with all the other arcs brought to bear upon the pro- 

 blem ; and then u and v (on which the sought-for semiaxes a 

 and b depend) are found from the condition that the sum of the 

 squares of all the above errors shall be an absolute minimum. 

 In this process Bessel considers x to be an independent variable. 

 It is in this that I think that he is wrong. I will now show 

 why. Assuming x to be an independent variable amounts, as 

 the process shows, to making the sum of all the corrections of 

 latitude of the several stations of the arc equal to zero. Hence 

 by this process x (the correction of the reference-station * of the 

 arc), with its sign changed, is equal to the sum of the relative 

 corrections of the other stations of the arc divided by the whole 

 number of the stations. No account whatever is taken of the 

 error which may be produced at the reference-station by local 

 attraction, which may be very large, so much so as to " over- 

 whelm *' other errors. Let t, f, t", ... be the deflections of 

 the plumbline (reckoned positive when to the north) at the suc- 

 cessive stations arising from local attraction. If the formulae 

 for calculating m, a, ft, . . . be examined (see ' Figure of the 

 Earth/ p. 127, or Ordnance Survey, p. 737), it will be seen 

 that the first term of m is the same as t'—t; for (Figure of the 

 Earth, p. 147, art. 149) the difference of deflection in the plane 

 of the meridian at two stations equals the difference between the 

 astronomical and mean amplitudes. Hence, denoting the re- 

 mainder of m by h, the corrections which Bessel applies to the 

 latitudes are 



x, t'—t + h + oiu + ftv + x,..., 



in which h } a, ft . . . are numerical quantities obtained from ob- 

 served latitudes and measured arcs. In these corrections local 

 attraction is taken account of in all of them except in that for 

 the reference-station. For that station a mere arbitrary chance 

 correction is applied ; and if the local attraction there be large, 

 the whole arc may be carried away by the " shifting " process, 

 which Captain Clarke so clearly describes, to an inordinate dis- 



* I have called this both the " reference-station " and the " standard 

 station " in my book. Captain Clarke observes that the latter is a term 

 not known in geodesy. I am therefore quite willing to drop it. I made 

 use of it for no special reason, and it is not desirable to multiply terms. 

 He calls it the "initial point" (British Ordnance Survey, p. 738). I will 

 retain " reference-station," as it so entirely represents the character of 

 the station ; for the position of the whole arc and its portions is deter- 

 mined by the observed latitude of that station, and by the observed ampli- 

 tudes of the subdivisions from that station. Hence the whole is referred 

 to that station. 



