MR. Cl. F. C. SEARLE ON PROBLEMS IN ELECTRIC CONVECTION. 
691 
(2.) A force Hr, where H is the magnetic force and r the volume-density of positive 
imaginary magnetic matter. 
These two forces follow from the ordinary laws of electrostatics and magnetism. 
(3.) A force YCB (Electromagnetic force), where € is the electric current density 
and B the magnetic induction. As far as I know no satisfactory proof of the formula 
has been given. Maxwell obtains this formula in § 602, vol. 2, of his ‘Electricity 
and Magnetism,’ but he assumes (practically) the result he is going to obtain, for he 
assumes that the force “ corresponding to the element ds,” actually acts on ds. The 
formula gives absolutely correct results when applied to find the force experienced by 
a complete circuit, and has besides the merit of simplicity. 
The expression can be deduced from Maxwell’s expression for the magnetic 
stresses in the field, but apart from the harmony which results when all the forces 
due to magnetic actions can be obtained from a single formula, no confirmation of its 
correctness is obtained, for the Maxwell stress was constructed so as to give the 
force YCB. 
(4.) A force — YGD (Magneto-electric force) where G is the rate of increase of the 
magnetic induction, or the “magnetic current,” and D is the electric displacement. 
This force, as Mr. Heaviside has remarked, can be deduced from the Maxwell 
electric stress provided that we assume that the stress is the same wdiether the 
electric force has a potential or not. The force has never been experimentally 
observed. 
Mechanical Stress between Two Systems. 
9. We shall now suppose the complete system to be made up of two separate systems 
of sources of disturbance, and will write down the force experienced by one of 
these systems due to the other. Since the sum of any number of solutions of the 
differential equations of the electromagnetic field is also a solution, it follows that 
if one of the systems of sources of disturbance gives rise by itself to a field charac¬ 
terized by E', H' and the other system gives rise by itself to the field E", H" and 
if E, H denote the field when both systems are present, then 
E = E' + E" H = H' + H". 
The force experienced by any portion of the medium per unit of volume is 
therefore 
(E'-f E") {p+p') + { H'+H") (t'+t")+Y (C'-f-C") (B'-f B")-V (G' + G") (D' + D"). 
If the force per unit volume which is due to the mutual action of the two systems 
be denoted by P, then 
P = E>" + Ey + Hr" + H'Y + YC'B" + YC"B' - YGD” - VG'D' . (63). 
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