244 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



not in the same diameter of the earth, nor equally distant from the centre in any 

 chord of the earth, the two axes of revolution of the sphere and tangential surface do 

 not coincide ; and hence their common intersection is not a plane, nor its trace on 

 the sphere, that is, the magnetic equator, a circle of any magnitude whatever. It is 

 therefore a curve of double curvature. The conclusion, therefore, deduced by Biot 

 from Humboldt's observations, and the conclusion deduced by Morlet from the 

 discussion of all the observations he could collect from authentic sources, are quite 

 consistent in this respect with the hypothesis of the duality of the poles. 



11. The discussion of any further cases of this problem need not be given here. In 

 a geometrical point of view the discussion would be interesting, and under that aspect 

 this paper would be incomplete without them ; but as they have no bearing upon the 

 main object of the present research, and are moreover so perfectly analogous to those 

 we have just given as to offer not the slightest ditficulty by the same method which 

 has been here employed, any further notice of them would be altogether superfluous, 

 and irrelevant to the purpose we have in view. 



12. It is to be remarked, however, that these results are true only on the hypothesis 

 of the forces in the poles being related by the equation F^ 4: P,, = 0. Under any 

 other condition than this the highest power of x would not disappear from equation 

 (59.) ; and hence the equation would be of an even order, and hence the branches of 

 the magnetic equator would be tivo at least, and always an even number. The ap- 

 parent singleness of branches furnished by observation is a strong argument in favour 

 of the duality of the poles and equality of their intensities ; but as the method by 

 which the magnetic equator has been laid down is far from satisfactory, too much 

 reliance should not be placed on this argument, decisive as it otherwise would 

 certainly be. 



13. The equation of the tangential surface and the equation of the sphere would 

 completely define that line considered in reference to rectangular coordinates ; that 

 is, in the usual manner of considering the equations of lines situated in space. In 

 the form of the equation of the locus of N on a meridian plane marked (60.), we 

 have only to write i/^ _j_ ^2 instead of y^^ and the result is the equation of the surface 

 referred to the axis of x and any other two axes at right angles to it and to one an- 

 other. The coordinates of the centre of the sphere referred to these axes, and the 

 radius of the sphere, being also given, we have its equation in the usual form, viz. 

 r2 = {x — a^)2 ^ (a^ — h,)^ + (2. — c^. By the transformation of coordinates we 

 can change the axes of reference to any given axes, as, for instance, to the polar 

 axis and the intersection of the equator by two rectangular meridians. In the next 

 place, to adapt the expression to the usual mode of denoting spherical position, 

 (latitude and longitude,) we must transform this into a polar equation, and put the 

 radius vector of the resulting equation constant and equal to the terrestrial radius. 

 The equation thus obtained will be one between the latitude and longitude of the 

 points which constitute the magnetic equator. 



