MAGNETIC SURVEY OF GREATT BRITAIN. 93 



for in such cases the curvature of the hues not being rapid, the 

 curve itself may, for a small portion of its extent, be confound- 

 ed with its tangent. It suggests perhaps the best mode of de- 

 termining with precision the empirical laws of the distribution 

 of terrestrial magnetism ; namely, by means of small groups of 

 observations, each of which will give, by this method, not a point 

 in the curve merely, but a portion of its tangent. 



The extent of the district in which this method is available 

 will, of course, vary with the curvature of the lines on the 

 earth's surface, becoming more and more limited as we approach 

 the poles. Where the flexure of the lines is rapid, and we 

 seek, nevertheless, to combine the observations scattered over 

 a moderately extensive tract of country, it becomes necessary 

 to obtain some means of pushing the approximation further. 



Such means I'eadily present themselves. Whatever be the 

 laws of distribution of magnetism on the surface of the earth, 

 it is manifest that the dip (or intensity) at any station is a func- 

 tion of its co-ordinates of position ; or that 



^ = F(a,^), 

 a and /3 denoting the co-ordinates of the station (in parts of 

 radius) referred to some neighbouring station as an origin. 

 Accordingly, 



+ I (^) 13' + &c. 



2 \ci i3y ^ 



the brackets denoting the particular values of the derived 

 functions, when a = 0, yS = 0. The quantities u and /3, in the 

 preceding equation, being small, we may push the approxima- 

 tion as far as we please, by including a greater number of terms 

 in the development. 



Let the co-ordinates of linear distance be denoted, as before, 

 by X and y, 



*' a y 

 r r 



r being the radius of the earth. Substituting these values in 

 the preceding equation, and making 



we have 



s; = L + M ^ -h N 2/ -1- P A'M- Q ^ 2/ + R 2/^ + &c. (7) 



