MAGNETIC FORCE IN IRELAND. 187 



III' Horizontal intensity in Limerick, the horizontal intensity 

 in Dublin being unity. 



If then x and y denote the horizontal intensities in Dublin and 

 Limerick, that in London being unity, observation gives 



x = "9380, y = -9460, | = 1-0038, 



and it is required to determine the most probable values of x and y. 

 To generalize this problem, let the mean results of observation be 

 a, b, c, and let their weights be A, B, C respectively ; so that we 

 have 



x - a - 0, weight = A, 



y - b = 0, J0, 



a and b being approximate values of x and y, let their true values 

 be x = a -\- x, y = b + S y, 



and let - = c . Then y - - *-^j- = c + a~ l (S y - c' Sar), the squares 



a x a + Sx 



and higher powers of the quantities S x and 8 y being neglected ; 

 so that the preceding equations may be written 



I x = 0, 8 y = 0, rr 1 (8y - e,S*0 + ^ - c = 0. 



These three equations are to be combined by the method of least 

 squares. "We find in this manner 



A$x- Cc/r l (a~ l $y - e<ar l %x + c, - c) = 0, 



-c =0; 



