CONCERNING TERRESTRIAL MAGNETISM. 229 



(cc" ^-^) (b" -J) ={(i"-^p)ia"--a) (31.) 



(u'"-a){b"'-b) = {p"'-J){a"'-^a) (32.) 



(«-_.^)(r_F) = (p'^-^)(«--a) (33.) 



This reduction is easily effected by subtracting the first from each of the others, in 

 which case we obtain equations of the first degree, giving each of the other three 

 the quantities a & a /3 in terms of the fourth, as of u. These substituted in any one 

 of the four equations give a quadratic equation involving a-, and hence we obtain 

 two values of a, and hence again of ^ of a^ and of ^. We should then obtain, by a 

 simple and direct process, the equations of the magnetic axis. 



The next inquiry is into the signification of this double result. Are there two mag- 

 netic axes which fulfill the condition ? If so, are they both occupied by magnets ? Or 

 if not, why is one to be selected in preference to the other ? Can they both belong to 

 even/ quadruple combination of the magnetic needle ? 



The last question may be answered at once. If they both belonged to all the com- 

 binations of the needles, then they must form two of the directrices of a rule surface, 

 to which the needles themselves were always tangents. The third directrix not being 

 yet fixed, there is no inconsistency in the conclusion thus derived ; for the needles 

 are at liberty to rest upon any magnetic surface, whatever be the number or intensity 

 of the poles, or whatever be the parameter which determines the particular stratum 

 of surface which corresponds to the place of observation on the sphere. There is 

 hence nothing to prevent their belonging to every position of the magnetic needle, 

 so far as we at present can discover from the conditions in their arbitrary form. 

 How far this is consistent with the particular data is another question, and will be 

 presently discussed. 



There is no necessity that they should be both occupied by magnets ; and it is at 

 once giving up the duality of the poles, and even their being situated in a right line, 

 to make such an hypothesis. They are both, it is true, solutions of the algebraical 

 problem which we have proposed ; but as the algebraical rarely includes all the con- 

 ditions of the physical problem, it is easy to suppose that one of these solutions may 

 be foreign to the inquiry, without violating our knowledge of the nature of the re- 

 lations subsisting between the algebraical and physical problem. To prove that it 

 actually is a foreign result must be subsequent to the determination of the particular 

 values of the coefficients of the quantities involved in the inquiry. All we can say at 

 present is, that there are two axes which fulfill the algebraical condition ; but as there 

 is only one which enters into the physical h5rpotheses, one of these two algebraical 

 axes must be rejected. We cannot, however, ascertain which, except by other con- 

 ditions than have yet been taken into the formula. For the present, then, we can 

 only compute them both, and take that which best answers to those other physical 

 conditions of which the algebraical problem has taken no account. 



