92 EIGHTH REPORT — 1838. 



L S {w) + Mt{2vx) +^'E (to y) = S {w ^), 



L S {w x) + M S (mj a:^) + N 2 (w x >/) = X{w x s:), (3) 



L 2 (m) y) + M 2 (^« a; y) + N S {iv ?f) =^{wys:); 



in which w denotes the loeight of the determination, and the 

 symhol S the sum of the n values of the quantities within the 

 brackets, n being the number of separate determinations. From 

 these equations, the most probable values of the three unknown 

 quantities, L, M, N, are obtained by elimination. 



If the point taken for the origin of the co-ordinates be that 

 for which 



2 (w x) =0, ^{loy) =0; 

 or be, as it were, the centre of gravity of the stations, the final 

 equations are reduced to 



L 2 {iv) = 2 {w s), 

 M 2 {iv x^) + I^t{tvx7/)=t (wx;:;), 

 M.t{tvxy) + N2 (w if) = 2 (toy:^). 

 The values of L, M, N being obtained, we may apply 

 the equation (2) either to determine the value of z, when x 

 and y are given, i. e., to deduce the most 2^robable value of the 

 dip for a given place, — or, conversely, to infer the relation of 

 X and y when s is given, i. e. to determine the eqtiation of the 

 line passing through all the points of given dip. In this latter 

 application let ;:; — L = K ; the equation of the line then is 



M a,' + N 2/ = K, (4) 



X and y being the co-ordinates, measured along the parallel of 

 latitude and the meridian respectively. On this supposition, 

 then, the isoclinal line is a right line ; the angle which it makes 

 with the meridian is 



ang (tan = — ^j J ; (5) 



and the increase of the dip corresponding to each geographi- 

 cal mile of distance, in a direction perpendicular to the line, is 



VM? + W. (6) 



In this mode of computation it is assumed, not only that the 

 portion of the earth over which the observations extend 

 may be treated as a plane surface, but also that the differ- 

 ences of dip (or intensity) are linear f\xnct.\o\\s of the differences 

 of latitude and longitude,— in other words, that the isoclinal 

 and isodynamic lines are straight. This supposition may be 

 safely made, where the district of observation, itself inconsider- 

 able in extent, is remote from the poles of dip or of intensity ; 



