194 Captain Clarke on the Figure of the Earth. 



being determined, x l} a? 2 , . . . are the local deflections at the 

 points P p P 2 . . . with reference to that spheroid. 



All this seems exceedingly simple; but Archdeacon Pratt, in 

 the third edition of his ' Treatise on Attractions, Laplace's Func- 

 tions, and the Figure of the Earth/ of which a notice appeared 

 in the last Number of the Philosophical Magazine, has obscured 

 it by attaching some special importance to certain points (one in 

 each arc) which he calls standard stations, a term unknown to 

 geodesists. For instance, St. Agnes is considered the standard 

 station of the English arc ; but on what grounds? because it is the 

 southernmost station ? If so, Saxaford has an equally good claim 

 to be considered the standard station because it is the northern- 

 most; and Greenwich Observatory might have a special claim. 

 But in reality the standard station is that with reference to which 

 the differences of latitude in each arc have been taken. The selec- 

 tion of the point in any arc with reference to which differences 

 of latitude are to be taken is merely a matter of arithmetical 

 convenience — nothing more : the southern extremity (in the 

 northern hemisphere) presents facilities; so does the northern; 

 but it is really quite immaterial, since the result, as is easily 

 shown, is quite independent of this consideration. Any station, 

 then, in an arc may be selected as the standard station in this 

 sense. 



If x v a? 2 , . . . be the corrections to the latitudes of (or local 

 attractions, or deflections at) the reference stations in each arc, 

 the corrections to the latitudes of the other stations will be 



1st arc. 



m x + U\U + /3 } v -\- x v . 



. . m' 1 + a' 1 M + /3\i> + # 1 , . 



2nd arc. 



nici + aju + fijv + scv . 



. . m'z -j- u' 2 u -f fi'cp + cT 2 , . 



3rd arc. 



m 3 + a 3 u + P 8 v + x 3 ,. 



. . m' 3 + a' 3 u + /3' 3 v -f- x 3 , . 



and the sum of the squares of these quantities 4-£ > 1 2 + <z 2 2 + • • • > 

 is to be a minimum; differentiating this quantity with reference 

 to u, v, x l} a? 2 , a? 3 . . . . , Archdeacon Pratt says, " This mode of 

 proceeding is, I conceive, erroneous, as I shall now endeavour to 

 show. The corrections x v # 2 . . . are not properly independent 

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

 produced by local attraction. " Then follows a demonstration in 

 which the point seems at first sight to be proved; but in the 

 demonstration it is tacitly assumed (see figure, page 259 of 

 No. 64, vol. xiii. of the Proceedings of the Royal Society, or 

 page 128 of Archdeacon Pratt's work, third edition) that the 

 southern extremity A, or reference-point, of the arc A B is a 

 fixed point. What is there to fix it ? Nothing is known as to 

 the actual position of the arc AB with reference to the semiaxes 

 of the meridian ellipse; what we know about the arc is simply 

 its length A B, and the inclination, to the axis of rotation, of the 



