^70 EEl-OET— 1888. 



But unless k is constant throughout our field, the four equations (12) and (14) 

 are mutually inconsistent ; from which it follows that our supposition of un- 

 changingness of electrification (§4) is not generally true. An interesting and 

 important practical conclusion is, that when currents are induced in any way, in a 

 solid composed of parts having different electric conductivities (pieces of copper 

 and lead, for example, fixed together in metallic contact), there must in general be 

 changing electrification over every interface between these parts. This conclusion 

 was not at first obvious to me : but it ought to be so by anyone approaching the 

 subject with mind undisturbed by mathematical formulas. 



8. Being thus warned off heterogeneousness until we come to consider changing 

 electrification and incomplete circuits, let us apply (10) to an infinite homogeneous 

 solid. As (8) holds through all space according to our supposition in § 4, and as 

 Kis constant, (13) must now hold through all space, and therefore ^ = 0, which 

 reduces (10) to — 



K at K M K at 



These equations express simply the known law of electro-magnetic induction. 

 Maxwell's equations (7) of § 783 of his ' Electricity and Magnetism,' become in this 

 case — 



K^'^«^^^1-)^=V--^- . ■ '■ . (150 



which cannot be right, I think, according to any conceivable hypothesis regarding 

 electric coudnctivity, whether of metals, or stones, or gums, or resins, or wax, or 

 shellac, or india-rubber, or gutta-percha, or glasses, or solid or liquid electrolytes ; 

 being, as seems to me, vitiated for complete circuits by the curious and ingenious, 

 but as seems to me not wholly tenable hypothesis which he introduces, in § 610, 

 for incomplete circuits. 



9. The hypothesis which I suggest for incomplete circuits, and consequently 

 varying electrification, is simply that the components of the electro-motive force 

 due to electro-magnetic induction are still 47r^"-duldt, &c. Thus, for the equa- 

 tions of motion, we have simply to keep equations (10) unchanged, while not 

 imposing (8), but instead of it taking 



.^,.(du^ch_^du:\d ,^^_ .... (16) 

 \d.v dy dz/ dt ^ 



■where p denotes 47r times the electric density at time t, and place {x, y, s), and 

 ' V ' denotes' the number of electrostatic units in the electro-magnetic unit of 

 electric quantity. This equation expresses that the electrification of which ■*■ is the 

 potential, diminislies and increases in any place according as electricity flows more 

 out than in or more in than out. We thus have four equations, (10) and (16), for 

 our four unknowns, ic, v, w, 'V ; and 1 find simple and natural solutions, with nothing 

 vague or difficult to understand, or to believe when understood, by their applica- 

 tion to practical problems, or to conceivable ideal problems, such as the transmission 

 of ordinary or telephonic signals along submarine telegraph conductors and land 

 lines, electric oscillations in a finite insulated conductor of any form, transference 

 of electricity through an infinite solid, &c. &c. This, however, does not prove my 

 hypothesis. Experiment is required for informing us as to the real electro-magnetic 

 eifects of incomplete circuits ; and, as Helmholtz has remarked, it is not easy to 

 imagine any kind of experiment which could decide between different hypotheses 

 which may occur to anyone trying to evolve out of his inner consciousness a theory 

 of the mutual force and induction between incomplete circuits. 



6. On the Transference of Electricity ivithin a Homogeneous Solid Conductor. 

 By Professor Sir William Thomson, LL.D., F.B.S. 



Adopting the notation and formulas of my previous paper, and taking p to 

 denote 47r times the electric density at time t, and place (x, y, z), we have — 



