6^- Prof. Thomson on the Mathematical Theonj of 



and sufficient for determining this distribution may (as can be 

 shown from Poisson's analysis) be expressed as follows. Let R 

 be the resultant force on a point P without C, and R' on a point 

 P' within C, due to the electrified surfaces A and B, and to the 

 imagined distribution on C. If P and P' be taken infinitely near 

 one another, and consequently each infinitely near the surface 

 of C, the component of 11' in the direction of the normal must 

 bear to the component of R in the same direction a constant 



ratio \~r) depending on the capacity for dielectric induction 



of the matter of C*. The components of R and R' in the 

 tangent plane will of course be equal and in the same direction, 

 and, if p be the intensity of the imagined distribution on the 

 surface of C, in the neighbourhood of P and P', the difference 

 of the normal components will be Airp, as is evident from 

 Coulomb's theorem, referred to above. 



Let us now suppose C to be a shell surrounding A, and let S 

 and S', its interior and exterior surfaces, be surfaces of equili- 

 brium in the system of forces due to the action of A and B, and 

 of the polarity of C. It may be shown that the same surfaces 

 S, S' would necessarily be surfaces of equilibrium, if C were 

 removed and the whole space were filled with air ; and conse- 

 quently, that the whole series of surfaces of equilibrium, com- 

 mencing with A and ending with B, will be the same in the two 

 cases. Hence the resultant force due to the excitation of the 

 dielectric C, or to the imagined distributions of electricity on S 

 and S' which produce it, on points within S or without S', must 

 be such as not to alter the distributions on A and B when the 

 quantity on A is given, and is therefore nothing. Accordingly, 

 let Q be the total force on a point indefinitely near S, and within 

 it ; Q' the total force on a point without S', but indefinitely near 

 it. Since the forces on points without S and within S' indefi^it 



* From this it follows, that, in the case of heat, C must be replaced by 

 a body whose conducting power is k times as great as that of the matter 

 occupying the remainder of the space between A and B. . - 



['fhe same demonstration, of course, is applicable to the influence of a 

 piece of soft iron, or other " paramagnetic" (i.e. substance of ferro-magnetic 

 mductive capacity), or to the reverse influence of a diamagnetic on the 

 magnetic force in any locality near a magnet in which it can be placed, and 

 shows that the lines of magnetic force will be altered by it precisely as the 

 lines of motion of heat in corresponding thermal circumstances would be 

 altered by introducing a body of greater or of less conducting power for 

 heat. Hence we see now strict is the foundation for an analogy on which 

 the amducting power of a maynetic medium for lines offeree may be spoken 

 of, and we have a perfect explanation of the condensing action of a para- 

 magnetic, and the repiilsive effect of a diamagnetic, upon the lines of force 

 of a magnetic field, wiiich have been described by Faradav. — (Exp. He- 

 searches, §§ 280/, 2808).] rii8«:io 



