Oct. ii, 1888] 



NA TURE 



57' 



„._, (dlt dv (fcc^ 

 \dx dy dz 



stones, or gums, or resins, or wax, or shell- lac, or gutta-percha, 

 or india-rubber, 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 electromotive due to electro-magnetic induc- 

 tion are still /\.irv'-dujdt, Sec. Thus for the equations of 

 motion we have simply to keep equations (10) unchanged, 

 while not imposing (8), but instead of it taking — 



')'*+• (,6) 



where "v" denotes the number of electrostatic units in the 

 electro-magnetic unit of electric quantity. This equation ex- 

 presses that the electrification of which "V is the potential 

 increases and diminishes in any place according as electricity 

 Hows more out than in, or more in than out. We thus have 

 four equations (10) and (16) for our four unknowns, u, v, 7V, ¥; 

 and I find simple and natural solutions with nothing vague, or 

 difficult to understand, or to believe when understood, by their 

 application to practical problems, or to conceivable ideal prob- 

 lems ; 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, trans- 

 ference of electricity through an infinite solid, &c. This, how- 

 ever, does not prove my hypothesis. Experiment is required 

 for informing us as to the real electro-magnetic effects of in- 

 complete 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. 



On the Transference of Electricity within a Homogeneous Solid 

 Conductor, by Sir William Thomson. — Adopting the notation 

 and formulas of my previous paper, and taking p to denote 4*- 

 times the electric density at time t, and place (x, y, z), we 

 have — 



V'-f = 



du dv d w \, 

 dx dy dz 



•(17) 



and, eliminating //, v, to, ¥ by this and (16) from (io), we find, 

 on the assumption of k constant — 



at 



vV = 



d'-p 



dt- 



»" a W 



(18) 



The settlement of boundary conditions, when a finite piece of 

 solid conductor is the subject, involves consideration of u, v, w, and 

 for it, therefore, equations (17) and (12) must be taken into ac- 

 count ; but when the subject is an infinite homogeneous solid, 

 which, for simplicity, we now suppose it to be, (18) suffices. It is 

 interesting and helpful to remark that this agrees with the equa- 

 tion for the density of a viscous elastic fluid, found from Stokes's 

 equations for sound in air with viscosity taken into account ; and 

 that the values of u, v, w, given by (17) and (10), when p has 

 been determined, agree with the velocity components of the 

 elastic fluid if the simple and natural enough supposition be 

 made that viscous resistance acts only against change of shape, 

 and not against change of volume without change of shape. 



For a type- solution assume — 



_ 7.1TX 2iry 2irz , . 



P = Ah"'/ f cos cos , cos .... (10} 



a b c y/ 



ind we find, by substitution in (18) — 



k "v" 2 



r - l# + -jT- = ° (20) 



where — 



v T >*/+(?* p* *) («) 



Hence, by solution of the quadratic (20) for q— 



[In the communication to the Section numerical illustrations 

 ■ non-oscillatory and of oscillatory discharge are given.] 



Five Applications of Fourier's Law of Diffusion, illustrated 

 by a Diagram of Curves with Absolute Numerical Values, by 

 Sir William Thomson. — (1) Motion of a viscous fluid ; (2) closed 

 electric currents within a homogeneous conductor ; ' (3) heat ; 

 (4) substances in solution ; (5) electric potential in the conductor 

 of a submarine cable when electro-magnetic inertia can be 

 neglected. 2 



r. Fourier's now well-known analysis of what he calls the 

 " linear motion of heat " is applicable to every case of diffusion 

 in which the substance concerned is in the same condition at all 

 points of any one plane parallel to a given plane. The differ- 

 ential equation of diffusion, 3 for the case of constant diffusivity, 

 k, is — 



dv _ d-v 

 It ~ K dx^ 



where v denotes the "quality" at time t and at distance x from 

 a fixed plane of reference. This equation, stated in words, is 

 as follows : — Rate of augmentation of the "quality" per unit 

 of time is equal to the diffusivity multiplied into the rate of 

 augmentation per unit of space of the "quality." 



The meaning of the word "quality" here depends on the 

 subject of the diffusion, which may be any one of the five cases 

 referred to in the title above. 



2. If the subject is motion of a viscous fluid, the "quality" 

 is any one of three components of the velocity, relative to rect- 

 angular rectilineal co-ordinates. But in order that Fourier's 

 ditfusional law may be applicable, we must either have the 

 motion very slow, according to a special definition of slowness ; 

 or the motion must be such that the velocity is the same for all 

 points in the same stream-line, and would continue to be steadily 

 so if viscosity were annulled at any instant. This condition is 

 satisfied in laminar flow, and more generally in every case in 

 which the stream-lines are parallel straight lines. It is also 

 satisfied in the still more general case of stream-lines coaxal 

 circles with velocity the same at all points at the same distance 

 from the axis. Our present illustration, however, is confined 

 to the case of laminar flow, to which Fourier's ditfusional laws 

 for what he calls "linear motion" (as explained above in § 1) 

 is obviously applicable without any limitation to the greatness 

 of the velocity in any part of the fluid considered (though with 

 conceivably a reservation in respect to the question of stability 4 ). 

 In this case the "quality" is simply fluid velocity. 



3. If the subject is electric current in a non-magnetic metal, 

 with stream-lines parallel straight lines, the "quality" is simply 

 current-density, that is to say, strength of current per unit of 

 area perpendicular to the current. The perfect mathemical 5 

 analogy between the electric motion thus defined, and the cor- 

 responding motion of a viscous fluid defined in § 2 was accentu- 

 ated by Mr. Oliver Heaviside in the Electrician, July 12, 1884 ; 

 and in the following words in the Philosophical Magazine for 

 1886, second half-year, p. 135: — "Water in a round pipe is 

 started from rest and set into a state of steady motion by the 

 sudden and continued application of a steady longitudinal drag- 

 ging or shearing force applied to its boundary. This analogue 

 is useful because everyone is familiar with the setting of water 

 in motion by friction on its boundary, transmitted inward by 

 viscosity." Mr. Heaviside well calls this analogue "useful." 

 It is, indeed, a very valuable analogy, not merely in respect to 

 philosophical consideration of electricity, ether, and ponderable 

 matter, but as facilitating many important estimates, particu- 



1 This subject is essentially the "electro-magnetic induction " of Henry 

 and Faraday. It is essentially different from the " diffusion of electricity" 

 through a solid investigated by Ohm in his celebrated paper " Die Galvan- 

 ische Kette maihematisch bearbeitet," Berlin, 1827; tranlated in Taylor's 

 "Scientific Memoirs," vol. ii. Part 8, "The Galvanic Circuit investigated 

 Mathematically," by Dr. G. S. Ohm. In Ohm's work electro-magnetic 

 induction is not taken into account, nor does any idea of an electric analogue 

 to inenia appear. The electromotive force considered is simply that due to 

 the difference of electrostatic potential in different parts of the circuit, un- 

 satisfactorily, and even not accurately, explained by wha', speaking in his 

 pre-Green;an time, he called ''the electroscopic force cf the body," and de- 

 fined or explained as "the force with which the electroscope is repelled or 

 attracted by the body ;" the electroscope being "a second movable body of 

 invariable electric condition." 



3 This subject belongs to the Ohmian electric diffusion pure and simple, 

 worked out by aid of Green's theory of the capacity of a Leyden jar (see 

 " Mathematical and Physical Paper," vol. ii. Art. 73*. 



3 See "Mathematical and Physical Papers," vol. ii. Art. 72. 



4 See "Stability of Fluid Mction," § 28, Philosophical Magazine, 

 August 1887. 



5 It is essentially a mathematical analogy only ; in the same sense as the 

 relation between the ''uniform motion of heat" and the mathematical 

 theory of electticity, which I gave in the Cambiidge Mathematical Journal 

 forty-six years ago, and which now consiitutes the first article of my 

 •' Electrostatics and Magnetism," is a merely mathematical analogy. 



