572 EEroBT— 1888. 



5. Electric potential in the conductor of a submarine caUe.' 



1. Fourier's now well-known analysis of what he calls the ' linear motion of 

 heat ' is applicable to every case of diHusion in which the substance concerned is in 

 the same condition at all points of any one plane parallel to a given plane. The 

 differential equation of diffusion,- for the case of constant diffuaivity «, is — 



dv cPo 



■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 tbe 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 live 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 rectangular rectilineal co-ordinates. 

 But in order that Fourier's diff'usional law may be applicable we must 

 either have the motion very slow, according to the special definition of slowness 

 in Article xcvii. below : 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 difiiisional 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 couceivablj' a reservation 

 in respect to the question of stability^). 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 ' qualify ' is simply current-density, that is to say, strength 

 of current per unitof area perpendicular to the current. The perfect mathematical* 

 analogy between the electric motion thus defined, and the corresponding motion of a 

 viscous fluid defined in § 2 was accentuated 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. 1 85 : ' 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 longi- 

 tudinal dragging or shearing force applied to its boicndat-y. 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 faci- 

 litating many important estimates, particularly some relating to telephonic con- 

 ductors and conductors for electric lighting on the' alternate-current' system. In 

 a short article to he included in volume iii. of my collected papers, which I hope 

 will soon be published, I intend to describe a generalisation, with, as will be 

 seen, a consequently essential modification of this analogy, by which it is extended 

 to include the mutual induction between conductors separated by air or other 



' This subject belongs to the Ohmian electric diflfusion pure and simple, worked 

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

 Physical Papers,' vol. ii., art. Ixxiii.). 



' See ' Mathematical and Physical Papers,' vol. ii., art. Ixxii. 



' See ' Stability of Fluid Motion,' § 28 Philosophical Magazine, August 1887. 



* 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 electricity, 

 which I gave in the Cambridge Mathematical Journal forty-six years ago, and which 

 now constitutes the first article of my ' Electrostatics and Magnetism,' is a merely 

 mathematical analog}-. 



