Captain Clarke on the Figure of the Earth. 193 



the indications of the black-bulb thermometer, as at present 

 constructed, are delusive, and they are especially so at great 

 elevations. 



I am, Gentlemen, 



Your obedient Servant, 

 Royal Institution, John TYNDALL. 



February 14, 1866. 



XXVIII. On Archdeacon Pratt's 'Figure of the Earth.' 

 By Captain A. R. Clarke, Royal Engineers, F.R.S., F.R.A.S.* 



IN the application of the method of least squares to the deter- 

 mination of the figure of the earth, there is this anomaly, 

 that the semiaxes are not determined by making the sum of the 

 squares of the errors of observation or measurement a minimum. 

 Of the existing uncertainty as to the figure of the earth, but a 

 small portion is due to errors of measurement or observation ; it 

 is mainly due to the circumstance that the earth is irregular, 

 and cannot be precisely represented by any ellipsoid or other 

 simple (algebraically speaking) surface. Imagine a spheroid 

 having its axis coincident with that of the earth's rotation, and 

 actually intersecting the mathematical surface of the earth in 

 various curves ; if P be any point in the former surface, p that in 

 which the normal at P meets the irregular surface, then, as far as 

 our present knowledge extends, and for values of the axes between 

 certain limits, Yp may be everywhere a very small quantity, and 

 we become aware of its existence only through its variations. 

 These appear as local displacements of the zenith, or, in other 

 words, discrepancies in the latitudes and longitudes of astrono- 

 mical stations. Now these latitude-discrepancies, or, as they 

 are otherwise expressed, deflections or local attractions, are in 

 their average magnitude very much larger than the probable 

 errors of the astronomical determinations of latitude, and indeed 

 overwhelm the errors of the geodetical operations. Moreover 

 the local deflections become, when we attempt to ascertain the 

 figure of the earth, inseparably mixed up with the errors of mea- 

 surement and observation. Hence to determine the figure of 

 the earth from a number of measured arcs and observed latitudes, 

 we have to find that spheroid which, when the measured arcs 

 are, as it were, applied to it, shall give the sum of the squares 

 of the deflections an absolute minimum ; in other words, correc- 

 tions x v x 2 , . . . have to be applied to the observed latitudes of 

 the points P p P 2 , . . . such as to bring them into accordance with 

 the spheroidal surface, while the sum of the squares of the quan- 

 tities x v <r 2 , . . . is to be an absolute minimum. The spheroid 

 * Communicated by the Author. 



