as obtained from Geodetic Data. 13 



must be a minimum. As the relative distances of a, b, c, d are 

 fixed, there is only one variable quantity, viz. the distance of a 

 from /; on this the other distances depend. The condition that 

 the sum of the squares should be a minimum leads to this other 

 condition, that the algebraical sum of the distances of a, b, c, d 

 from /, m, n, r should equal zero. This will give the value of la, 

 it is a constant but unknown quantity depending upon the form 

 of the mean ellipse ; let it for the present be called z. The form 

 of z will be calculated in the next paragraph. The arc abed 

 thus defined is the portion of the mean ellipse which represents 

 the measured arc ASCD. 



5. I will now calculate the form of z. Let L, M, N, R be 

 the points on the mean ellipse at which the normals are parallel 

 to the plumbline at A, B, C, D. Then the arcs hi, Mm, «N, 

 r R represent the local deflections at A, B, C, D*. Let them be 

 called t, t 1 , t n , t" 1 , the deflection being reckoned positive when 

 it is to the north. Then la = ha — t. Also ha being the in- 

 crement added to the observed latitude of A which gives the 

 latitude of a, then M b, the corresponding increment to the ob- 

 served latitude of B to obtain the latitude of b, will, by the for- 

 mulae of paragraph 3, be ha-t-m + ex!'\J -\- (3'V, 



.-. mb — Mb— Mm = Lfl + w' + «'U + /3'V- 1 ! . 



In the same way the distances of c and d from n and r are 



La + m"+«"U + /3"V-/", 

 La + m ll, + ct'"V + ^"V-t flf . 



Let (m) be a symbol which represents all the ms, and so of 

 the other quantities. Then the condition deduced in the last 

 paragraph, that the algebraical sum of the distances of all the 

 points a, b, c, d from /, m, n, r is to be zero, gives 



0=4La+ (m) +(«)U+ (/9)V- {t) ; 

 or generally, if i represents the number of stations on the arc, 



0=i.La+[m) + {a)V+(l3)Y-(t), 

 and 



ha = L/+ la = t -f z ; 



... ~_W t -W + («)XJ+(/3)V 



i i 



6. Having found z, I proceed to find the remaining part of x. 

 Draw AX, the normal from A on the variable ellipse, and sup- 

 pose Xu equals the unknown constant z. Then x, the correc- 



* Of course in the diagram small quantities are enormously exaggerated 

 to make them visible to the eye. 



