222 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



that have been made between its supposed results and the observations made on the 

 needle at certain places, and especially respecting the Halleyan lines, and Hansteen's 

 poles of greatest intensity, have caused the hypothesis to be rejected by many persons, 

 who, if they had looked more closely into the question, could not have failed to dis- 

 cover that their conclusions were altogether premature, and probably erroneous. I 

 speak now of the broad features of the phenomena compared with a popular rather 

 than a calculated series of deductions from the hypothesis. Whether, however, when 

 the results come to be more closely tested by an appeal to the numerical values of 

 the quantities in question, the same accordance would be found, is a question alto- 

 gether different : and it is one which we are not at present in a condition, for want of 

 numerical data, upon which to offer a distinct opinion, much less are we entitled to 

 express a positive decision concerning it. 



The present series of papers is chiefly intended to deduce the mathematical conse- 

 quences of the theory of two poles situated arbitrarily within the earth, and especially 

 to investigate the singular points and lines which result from the intersection of the 

 earth's surface with other surfaces related to the magnetic poles. Amongst these are 

 the magnetic equator, the points at which the needle is vertical, the lines of equal 

 dip, the Halleyan lines, the isodynamic lines, and the Hansteen poles. If it shall 

 appear in the course of these developments that the general features of all these are 

 roughly represented by the hypothesis of two poles, then it will be a strong argument 

 in favour of that simple theory ; but should a result, in any one of these cases, 

 follow from that hypothesis which is very decidedly opposed to the corresponding- 

 observed phenomena, we shall be compelled, if our observations are authentic and to 

 be depended on as unaffected by an extreme degree of foreign influence, to abandon 

 it altogether. 



Our hope of being able to separate the disturbing from the primary forces must 

 depend altogether upon their relative quantities. The success of astronomical re- 

 search has hinged wholly on the relative smallness of the disturbances in comparison 

 with the primary forces that govern the motions of a planet : and if the same order 

 of magnitude should exist in the magnetic forces in question, the same success will, 

 there is every reason to hope, follow in due time. If not, the research should be 

 placed at once amongst the desperata. However, till some method is discovered of 

 ascertaining whether such is the case or not, we should leave no effort untried either 

 to accomplish the proposed object, or to render manifest its impracticability. With 

 that view the present investigations, which are conducted in a manner altogether un- 

 tried before by any one, are offered to the attention of geometers, as being calculated, 

 besides exhibiting the general consequences of the dual hypothesis, in some degree to 

 point out where we should look for the influence of foreign forces, and especially 

 showing that in reference to one great circle, the want of symmetry in the results at 

 positions taken symmetrically with respect to it, gives us great cause to suspect the 

 action of such foreign forces. 



