Captain Clarke on the Figure of the Earth. 195 



tangents at A and B. These alone are fixed ; so long, then, as 

 the curve moves parallel to itself, its position is arbitrary. In 

 fact its position is determined by the best way in which it can 

 accommodate itself to the hypothetical ellipse, changing with every 

 variation of the variable ellipse. 



Imagine an arc of, say, four astronomical stations A, B, C, D ; 

 then we know the lengths AB, AC, AD, and we know the 

 directions of the tangents at A, B, C, and D; this is all. As 

 for the local attractions at these astronomical stations, they can- 

 not in any case be pronounced upon even with carefully con- 

 toured plans of the ground ; for the effect of irregularities which 

 exist in the density of the crust of the earth cannot be ascer- 

 tained. The local attractions at A, B, C, D can only be inferred 

 by comparing this arc with received elements of the figure of 

 the earth. Suppose that only one allowable value of the semiaxes 

 exists, and let the ellipse be described ; the curve A B C D 

 must then (retaining its fixed direction) be shifted about until it 

 coincides as closely as possible with some portion of the ellipse. If 

 it can be so managed that A B C D can be placed against other 

 points a, b, c, d of the ellipse so that the tangents at A and a, B 

 and b, C and c, D and d are respectively parallel, then we con- 

 clude that there exists no local attraction at any of the stations, 

 or that it is equal and in the same direction at all the points — 

 a less probable hypothesis. ' If parallel tangents cannot be found 

 on the ellipse, then the arc must be so adjusted that the sum 

 of the positive angles included by the corresponding tangents 

 shall be equal to the sum of the negative angles. In other words, 

 the algebraic sum of the apparent residual local attractions at 

 A, B, C, D must be zero. This is merely the principle of the 

 arithmetic mean. Now make a different supposition as to the 

 semiaxes, and we have a different ellipse. Fitting the same arc 

 to this as before, we get another system of most probable deflec- 

 tions differing from the former, but whose sum is also zero. 



For every hypothesis as to the figure of the earth there exists 

 then a corresponding most probable system of deflections for the 

 astronomical stations on any arc. But Archdeacon Pratt immove- 

 ably fixes the southern point or standard station of his arc, and so 

 falls into error. One of the erroneous results of this method of 

 calculation may be quoted: "for example, a deflection of the 

 plumbline of only 5" at the standard station (St. Agnes) of the 

 Anglo- Gallic arc would introduce a correction of about one mile 

 to the length of the semi- major axis, and more than three miles 

 to the semi- minor axis." Wow if there had been any such dis- 

 turbance of the plumbline at St. Agnes, the method of least 

 squares would without fail have found it out, and the correspond- 

 ing error introduced in the semiaxes would have been only a few 



