THE THEORY OF TERRESTRIAL MAGNETISM. 443 



These equations give the values of x and y for the point required. 



Also the values of X and Y for the point required are given by the 

 equations 



A \ _1_ I _ . 1 v _I_ I \ tl -I- - I I T^ -I I I tfi -I l I wit 



' \ 7 I W i I 7 / / l^ i- l la/*** T^ ,r I 7 < / '/ ^^ I "7 7 / wW. 



\rtas/o \%/o 2\da?/o 2\dy-/ J \dxdy/ ' 

 and 



/c/r\ /dy\ , i /epn i /^r\ /d 2 r\ 



, o;+l- W-4--I- - a; 2 + - ^^ w 2 -4-l^ ^- 



/a T 2 \dtfj, 9 r \dxdy ) * 



Y 



and tan 8 = -^ gives the value of the declination. 



Let 



be the values of the function ( 1, 1) (0, 1) (1, 1) 



(-1, 0) (0, 0) (I, 0) 

 (-!,-!) (0, -1) (1, -1) 



X to be determined at the 



points 

 Then taking differences along x, we have 



7 "V /* ,3 



, ,. iii Q>JL T a 



c d, r e and the mean -= = ; 



dx 2 



and taking differences along y, 



j j dX bh 



b e, e li and the mean -7- = - 



dy 2 



The second differences are 



d'X _ d^+f-Ze d*X _ , 



dx* = ~~2 ' Utf ~~ ~H ' 



also taking all the successive first differences with respect to x, 



b a, c b, 

 e-d, f-e, 

 i h, h y, 

 and the differences of these with respect to y, 



b a e + d, c b f+ e, 

 e d i + h, f e h + g, 

 and taking the mean of these, we have 



d*X 1 , 



Hence we have 



f-d b-h d+f-2e , b + h-2e 

 =e + J --x+--+- -*+- '- 



Similarly the value of Y at the point (x, y) may be determined. 



-xy. 



562 



