232 MR. DAVIES'S GEOMETRICAL INVESTIGATIONS 



I have not thought it necessary to add any further discussion of the question in the 

 present stage of my investigations, in as much as till the results can be ensured as un- 

 affected by extraneous sources of error, all methods must alike be useless, since they 

 are alike dependent upon data that are at least unsatisfactory if not erroneous. There 

 is some reason, however, to hope, since the attention of the scientific world is now so 

 intensely turned to researches of this nature, that there will at length be discovered 

 some methods of observing which shall be free from this class of errors. However, 

 till this is done, it would be useless to attempt the discussion of the present or any 

 other method of mathematical investigation into its numerical details : and the ut- 

 most we can now perform is to lay down methods of investigation by which, when 

 satisfactory experimental data are obtained, the question may be brought to a deci- 

 sive test at once. 



I should also state here, that in consequence of the great labour attending the cal- 

 culations of the axes, I have been led to examine the method of construction by de- 

 scriptive geometry, (especially on account of the facility and the considerable degree 

 of certainty which may be attached to its solutions,) of the problem of describing a 

 line which shall rest upon four given lines. In any case where the data are so uncer- 

 tain as the present, such a method is sufficiently accurate, since in very few cases will 

 the errors of construction be probably near so great as the errors in the data them- 

 selves. The geometrical problem itself is not in practice so simple as could be desired, 

 at least by any method yet made public, but still it offers far greater facilities than 

 the algebraical one. It has, moreover, one important advantage which the algebra- 

 ical has not, viz. the ready and visual exhibition of those cases which are unfit for 

 this method, or those in which a small error in the data will greatly increase in the 

 result. It should hence be always used before the algebraical. No doubt the alge- 

 braical method may be rendered subservient to the same purpose, but then the me- 

 thod is intolerably operose, and hence, practically, almost useless. 



By employing such constructions, I find that there is a greater degree of approxi- 

 mation in the few magnetic axes which I have determined by them from existing data 

 than appears compatible with any other theory of the constitution of the terrestrial 

 magnet than that which considers the magnetic force situated in two isolated centres 

 or poles. By this I would not be understood to say that the approximation is close, 

 but simply that, in comparison with all the positions which lines may take, there 

 seems to be one region of space, in reference to the coordinate planes or planes of 

 projection, into which they dispose themselves, but dispose themselves very irregu- 

 larly in it. 



As I expect to be favoured, by the kindness of my distinguished colleague Mr. 

 Christie, with the results of the observations of the late lamented Captain Foster, 

 I shall probably resume this branch of the subject at an early period ; and hence any 

 further details respecting these constructions which may appear necessary will be 

 with more propriety included in a future than in the present paper. 



