12 Archdeacon Pratt on the Figure of the Earth, 



ellipticity is nearly 1-^-300, let a and b be expressed by the 

 formula? 



¥ = 650 ( l - WO + ^o)^890000 feet, 

 6= 30o( 1+ 5o)' 



and 



w 



here u and v are small quantities determining the dimensions 

 of the ellipse. Suppose afiySis the portion of this ellipse which 

 represents the arc A BCD, the mutual distances of «,/3,y,8 being 

 the same as of A, B, C, D ; and let K be the point at which the 

 normal is parallel to the plumbline at A, and let Ku = x. x, 

 measured in parts of a terrestrial degree, is evidently the small 

 angle which must be added to the observed latitude of A to ob- 

 tain the latitude of a. The corresponding angles which must be 

 added to the observed latitudes of B, C, D are (the well-known 

 formulae) 



m! + a!u + @'v + oo, ml' + ct"u + @"v + x, .... 



where m!, a!, {3 f , . . . are numerical quantities depending upon ob- 

 served latitudes and measured lengths of arcs. Those values of 

 u and v which will make the sum of the squares of these cor- 

 rections a minimum are those which give the ellipse which most 

 nearly coincides with the arc in question. This is called the 

 Mean Ellipse. Let U and V be the values of u and v which cor- 

 respond to this ellipse. 



4. Before we apply the principle of least squares, as described 

 in the last paragraph, it is necessary to determine the form of x> 

 which is as yet an arbitrary quantity. Suppose «, b> c, d are the 

 positions which a, /3, 7, h assume when the variable ellipse coin- 

 cides with the mean ellipse, these points being at the same mu- 

 tual distances measured along the ellipse as the stations A, B, 

 C, D are from each other. The correction oc must be so deter- 

 mined as to make abed suitably represent the position of the 

 geodetic arc A B C D when referred to the mean ellipse. 



Draw A I, B m, C n, D r normals to the mean ellipse ; then, if 

 the points /, m, n, r are at the same distance from each other as 

 A, B, C, D are, the arc Imnr will properly represent ABCD, 

 as it will be its direct projection upon the mean ellipse. This 

 will not, however, be generally the case. The points a, b, c, d 

 must therefore be as near to /, m, n, r as possible ; that is, the 

 sum of the squares of the distances of a, b, c, d from /, m, n, r 



