as obtained from Geodetic Data. 15 



of the squares of the corrections of latitudes which is to be made 

 a minimum is 



x z+(m' + u , u + (3 , v + xy 2 +{m" + a"u-{-@"v + x)' 2 -t- .. . 



Differentiating first with respect to u and next to v, and then 

 putting U and V for u and v, we have 



0=(m' + a'U+/S / V + ( r)a , 4-K + « / 'U-f/3' / V + ^)a"+ ... 



= ff»+ (ro' + a'U + 0V + x){P + n) + (ro" + a"U +/3"V + #) (£" + w) + ... 



or, introducing symbols, 



0= M+ (a 2 )U+ (a/3)V + a?(a), 



= (ro + (/3)> + (»z/3) + («/5)U + (/3 2 )V 

 + w(m) -f w(a)U + n(/3)V; 

 or if we substitute for x y first putting V for v, 



o =w _^) + { (a2) -(f 2 }u + {( a/3 )-H^}v + (?) W) 

 0=N 8)-<=P + { W) -(f}u + [ m -m s Y + ( n+ (|)) W . 



These agree with the equations obtained by BessePs method 

 (see Captain Clarke's calculation, Ordnance Survey Volume, p. 

 738), with the exception of the terms which depend upon (t), or 

 the sum of the local deflections at all the stations of the arc. 



8. It will be seen, as might have been anticipated, that taking 

 into consideration the effect of local attraction introduces an 

 element of considerable uncertainty, as there is no known method 

 of finding what that sum is. The only way that I can see of 

 surmounting this difficulty is that which I pursued in the paper 

 published in No. 64 of the Royal Society's ' Proceedings/ viz. 

 finding the axes of the three long arcs (the Anglo-Gallic, the 

 Russian, and the Indian) by means of the above equations, in 

 terms of the three unknown sums (t x ), (t 2 ), (t 3 ), making these 

 axes the same in the three ellipses (which process will give four 

 equations connecting these three unknown), and then by the 

 method of least squares finding the values of (tj, {t 2 ) } (/ 3 ) which 

 best suit the equations— which values, when they are thus found, 

 will determine the numerical values of the mean axes. 



9. If two or more arcs are combined to find the axes, then, 2 

 (being a symbol meaning that the sum of all such quantities as 

 are placed after it is to be taken) passing from arc to arc, the 



