[ 332 ] 



XLI. On the Self-induction of Wires, — Part III. 

 By Oliver Heavislde*. 



THE subject of the decomposition of an arbitrary function 

 into the sum of functions of special types has many 

 fascinations. No student of mathematical physics, if he possess 

 any soul at all, can fail to recognize the poetry that pervades 

 this branch of mathematics. The great work of Fourier is 

 full of it, although there only the mere fringe of the subject 

 is reached. For that very reason, and because the solutions 

 can be fully realized, the poetry is more plainly evident than 

 in cases of greater complexity. Another remarkable thing to 

 be observed is the way the principle of conservation of energy 

 and its transfer, or the equation of activity, governs the whole 

 subject, in dynamical applications, as regards the possibility 

 of effecting certain expansions, the forms of the functions in- 

 volved, the manner of effecting the expansions, and the possible 

 nature of the " terminal conditions " which may be imposed. 



Special proofs of the possibility of certain expansions are 

 sometimes very vexatious. They are frequently long, com- 

 plex, difficult to follow, unconvincing, and, after all, quite 

 special ; whilst there are infinite numbers of functions equally 

 deserving. Something of a quite general nature is clearly 

 wanted, and simple in its generality, to cover the whole field. 

 This will, I believe, be ultimately found in the principle of 

 energy, at least as regards the functions of mathematical 

 physics. But in the present place only a small part of the 

 question will be touched upon, with special reference to the 

 physical problem of the propagation of electromagnetic dis- 

 turbances through a dielectric tube, bounded by conductors. 



It will be, perhaps, in the recollection of some readers that 

 Professor Sylvester, a few years since, in the course of his 

 learned paper on the Bipotential, poked fun at Professor 

 Maxwell for having, in his investigation of the conjugate pro- 

 perties possessed by complete spherical-surface harmonics, 

 made use of Green's Theorem concerning the mutual energy 

 of two electrified systems. He said (in effect, for the quota- 

 tion is from memory) that one might as well prove the rule 

 of three by the laws of hydrostatics, or something similar to 

 that. In the second edition of his treatise, Prof. Maxwell 

 made some remarks that appear to be meant for a reply to 

 this ; to the effect that although names, involving physical 

 ideas, are given to certain quantities, yet as the reasoning is 

 purely mathematical, the physicist has a right to assist himself 

 by the physical ideas. 



* Communicated by the Author. 



