Local Attraction on Geodetic Operations. 



361 



Therefore the sum of the squares of errors, which is to he differentiated 

 •with respect to u and v to obtain a minimum, is 



~ V) + (m, + + ftv + - V) + ^,)' 



{n,{v-M) 4- t,y + (m, + + (^,v + n,{v-y) + t,f 



+ (m'2+a>+/3>+w,(2;-V) + ^2)'H- • • - . 

 + = a minimum. 



Let U and V be the values of u and v which belong to the mean ellipse. 

 These values, then, must be put for u and v in the two equations produced 

 by differentiating the above with respect to u and v, We have 



a,K+a,U+ftV+^,) + a\(m',+<U+/3',V+0+ • • • 

 + a,(m,+ a,U+/3,V+g + a',(m>a',U+/3',V+g + . . . 

 -f =0; 



and 



'^A^ (ft + «0K + a,U + ftV4-^0 + (/3\ + w,)(m\ + a',U+/3',V+^,)+ . . . 



+ (/3, + n,) {m, + aJJ + /3,V + 1,) -f (/3', + {m', + a',U + /3',V + + . . . 

 + =0. 



Let (m) be a symbol representing the sum of all the m's appertaining 

 to the divisions of the same Arc ; and let S(m) represent the sum of all 

 these sums for all the Arcs ; and similarly for other quantities besides m. 

 Then the above equations become 



S(ma) +2(a2) U + S(a/3) V + S?(a) =0 

 and 2(m/3)-l +S(a/3) 1 +S(/30 1 V + 2^(/3)1 



4-Sw(m)J 4-S«(a) J +2«(/3)J J 



2 being the number of stations on the representative Arc. 



The numerical quantities involved in the first two lines of these equa- 

 tions have been already calculated in the article on the Figure of the Earth 

 in the British Ordnance Survey Volume, from which I borrow the results 

 in Table II. on the following page. The quantities involving n are calcu- 

 lated in Table I., and the results inserted in Table II. with the others. 



