INTRODUCTION AND RESULTS. 
249 
“ The position of a station of observation is thus determined by a distance in geogra- 
phical miles measured along the meridian which passes through the central station, 
and a distance in geographical miles measured along that parallel of latitude which 
passes through the station of observation. 
“Let z be the magnetic element whose value is to be determined, which in this case 
is the total intensity, and let sq, z 2 , &c. represent the values observed at ,q, s 2 , &c. 
“ Assume that z may be expressed by the formula 
z = ax 2 + bxy -f- cy 2 -\-dx-\-ey -f- f. 
“ This amounts to assuming that the isodynamic curves may be represented by a 
series of similar and similarly situated concentric ellipses, on a plane projection of the 
sphere, in which parallels of latitude are represented by equidistant horizontal straight 
lines, the meridian passing through the central station by a vertical straight line, and 
the other meridians by curved lines which all intersect in the projection of the poles, 
and each of which intersects the projection of the equator at right angles at a distance 
from the central meridian proportional to the difference of its longitude and that of 
the central meridian, (and equal to the projection of a corresponding number of 
degrees of latitude,) and intersects the other parallels of latitude at distances from the 
central meridian, which are to the last-mentioned distance as the cosines of the lati- 
tude of the respective parallels of latitude are to unity. 
“ Each station gives an equation of the form 
* 1 = ax 2 + bx l y l + cy 1 2 +dx l +ey l +f 
z 2 = ax 2 + bx,yj 2 + cy 2 + dx 2 + ey 2 +/ 
2„= ax 2 + bx n y n + cy 2 + dx n + ey n +/. 
“ These equations being properly weighted are then combined by the method of least 
squares, and the values of the constants a, b, c, d, e and f are to be determined by 
elimination. One test of the applicability of this method is, that the resulting curves 
should be ellipses and not hyperbolas; a and c must therefore have the same signs, 
and 4ac must be greater than b 2 . 
“ Having determined the constants a, b , &c., the coordinates x, y of the common 
centre of the" ellipses, which is also the place of the greatest intensity, are given by 
the equations 
2ax-f by-\-d=0, 
which give 
be — 2 cd 
X 4 ac—b 2 
&r+2cy-{-e=0, 
■ • • ( 1 .) 
- bd — 2ae 
^ 4 ac—b 2 
( 2 .) 
The maximum value of the intensity is 
*=f+ 
bde—cd^—ae 1 
4.ac—b‘ i 
(3.) 
