Action of a Galvanic Coil on an external small Magnet. 443 



In the particular case when c is diminished without limit, this 

 becomes 



2iroc *' „ » 

 r (** + ««)* 

 It will hence be seen that by this reasoning the inverse cube 

 of the distance is arrived at in the same manner as by that indi- 

 cated in arts. 21 and 22 relative to Gauss's empirical theory of 

 magnetism. 



In the same work (§ 546 (III.)) the formula for "the poten- 

 tial U of a disk of infinitely small thickness c with positive and 



negative matter of surface-density - on its two sides," as obtained 



by inference from the above expression^ is 



I 2irp(l x — A 



r V {x* + a*)i/ ;. 



the point Q acted upon being situated on the perpendicular to 

 the disk through its centre C. For cases in which Q is at any 

 distance r from C greater than a, and QC makes an angle 6 

 with the perpendicular, the formula for U is found to be 



the factors Q v Q g , &c. being functions of which satisfy the 

 equality 



(1 - 2e cos 6 + e 2 ) '-* = 1 + QjC + Q 2 e 2 + Q 3 e 3 -f &c. 

 The formula thus obtained for the potential U is the one used by 

 Mr. Stuart in the calculations already referred to in arts. 51 and 

 54. If r be considerably greater than a, the first and principal 



term of tt- varies inversely as r 3 ; and, as before, this result de- 

 pends on the hypothesis of positive and negative magnetic matter 

 on the two sides of the disk. 



77. Having given the foregoing explanations, I have now only 

 a few additipnal remarks to make. Whilst the hypothesis by 

 means of which the law of the inverse cube has been experimen- 

 tally educed, as respects both magnetic and galvanic action, is 

 entirely empirical, arising out of no antecedent reasoning, in the 

 hydrodynamical theory, on the contrary, that law is arrived at by 

 mathematical deduction from initial principles ; and this is true 

 both in the theory of magnetic action and in that of galvanic 

 action, inasmuch as the expressions for the two parts, direct and 

 indirect, which the latter action was found to consist of, both 

 vary, in the first and principal term, according to the law of the 

 inverse cube. Thus, since the hypothesis that a complete circular 



