CONCERNING TERRESTRIAL MAGNETISM. 231 



number, will serve as tests of the truth of the dual hypothesis with the poles of 

 equal intensity. 



It will be the most convenient method of taking the point O, to assume for it the 

 intersection of the magnetic axis by the perpendicular from the centre of the sphere. 

 Having, then, the distances of the poles from this point, and the equations of the line 

 in which they lie, we can easily determine their coordinates, the great object of our 

 inquiry. 



It can be no objection to this process that it necessarily requires the solution of 

 equations of a high order, since it is only the solution of it in the case of given nu- 

 meral coefficients, and not with literal coefficients. The method of effigcting these 

 solutions with rapidity and precision is now well known, and need not here be dwelt 

 upon. We shall have occasion hereafter to employ them in the numerical solution of 

 this special problem*. 



IX. — If we suppose the intensities unequal, we can assume their ratios to be that 

 of F, to F^;, or R. The relation upon which (5.), (6.), (70? ... are founded no longer holds 

 good in this case. Nevertheless, by reference to (XIV.) we see that the difference is 

 only in the numeral coefficients of the equations, and not in the form or the number of 

 terms, or in any circumstance that alters in the slightest degree the labour or the dif- 

 ficulty of the actual solution. We have however, in consequence of the new quantity 

 R which is thus introduced, to employ one equation more derived from observation-)-, 

 and one only. Hence still in this case, too, four observations are sufficient, not only 

 for the determination of the actual position of the poles, but also to furnish a test of 

 the accuracy of the hypothesis. As a method, then, this also is complete, and the 

 problem is fully brought within the reach of known and familiar operations. 



X. — As a specimen of the method of computing the equations of the magnetic 

 needle, I have given calculations for Chamisso Island, Valparaiso, Paramatta, Port 

 Bowen, Paris, and Boat Island ; and that the whole process may be distinctly seen, 

 I have also given the equations of the magnetic axis itself as deduced from the equa- 

 tions of the first four needles, and a comparison of the result with the Paris needle. 

 That result is not very favourable to the theory, provided the observations themselves 

 are considered trustworthy. But since those philosophers who have had most ex- 

 perience in the use of magnetical instruments, and especially of the dipping-needle, 

 are most strongly convinced that there are errors attached to all our present instru^ 

 merits and modes of observation whose amount vitiates any result obtained by them, 



* I refer, of course, to Mr. Horner's method, published in the Philosophical Transactions for 1819, and in 

 the fifth volume of Professor Leybourn's Mathematical Repository. It is unnecessary to add, that all the 

 f^ective methods of solution of algebraical equations that have since appeared have been but imitations of 



Ax. Horner's, however much the notation and /orm of the reasoning employed in them may differ from his. 



t Or if we seek to determine the actual value of F^ and F^^ we shall want all the four complete observations. 



