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L XVIII. On the Mathematical Relations of the Magnetic Field. 

 By a. H. Livens *. 



IT has already f been noticed that the usual conception of 

 the mechanical relations of the magnetic field, which 

 is based on the assumption that the magnetic force is the 

 fundamental vector, is somewhat misleading, if not entirely 

 erroneous, and that in order to obtain a more consistent theory 

 of these relations it is necessary to invert the usual order of 

 things and take the magnetic induction as the fundamental 

 force vector and the magnetic force as the induced vector. 

 The object of the present note is to present in a concise form 

 the results obtained by following through the usual argu- 

 ments but starting with the more consistent fundamental 

 conception. 



In all physical theories it is usual to describe the relation 

 between the inducing force and the induced effect by saying 

 that the latter is an explicit function of the former. The 

 inversion of such a description, which would give the inducing 

 force as an explicit function of the induced effect, may not 

 necessarily be incorrect, but it is at least inexpedient, as 

 it is very liable, as in the present case, to lead to a serious 

 misapprehension of the physical processes involved. 



The mathematical relations of the magnetic field are 

 ahvays expressed in terms of three vectors : (i.) the magnetic 

 force H, which is defined at any point of the body as the 

 vectorial ratio to a small magnetic pole of the force on it, 

 if placed there ; (ii.) the magnetic jjolarization intensity I, 

 which is the effective resultant bi-polar moment per unit 

 volume of the medium at each place ; and (iii.) the magnetic 

 induction vector B, which in the elementary theory is best 

 regarded as a composite vector induced by the force H, and 

 such that 



B = H + 4ttI. 



In the more consistent form of these relations the magnetic 

 induction B is taken as the fundamental force vector and 

 the magnetic force H as the induced vector, and then it is 

 better to write 



H = B-4ttI. 



* Communicated by the Author. 



t The complete development of the usual argument, with complete 

 references, will be found in my book ' The Theorv of Electricity ' 

 (C. U. Press, 1918). 



