110 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



between two fluids, but on account of the great mathematical difficulty 

 restricted the investigation at first to the .simplest case of the motion 

 of a rectilinear wave line which proceeds unchanged in form and with 

 a constant velocity along the unlimited bounding surface between two 

 fluids of different density. As a level water surface over which blows 

 a wind of constant strength is in a condition of neutral equilibrium, and 

 thus readily permits the creation of water waves, so it is with layers of 

 air of different densities, except that here the phenomenon progresses 

 on a vastly greater scale. Helinholtz, therefore, investigated the rela- 

 tions of energy and its division between air and water, and was led to 

 very general mechanical speculations whose consideration will form the 

 conclusion of our account. Very interesting but very difficult deduc- 

 tions were made, which established the difference between stable and 

 labile equilibrium, and just as, long before, the condition of stable 

 equilibrium for stationary bodies was found to be a minimum of the 

 potential energy so for stationary waves with constant velocity-poten- 

 tial the condition of stable equilibrium was found to correspond to the 

 minimum of energy. 



I turn now to sketch the last great category of his mathematical 

 physical labors, which led him finally to discoveries of the greatest sig- 

 nificance for the principles of mechanics. I refer to his investigations in 

 electricity, which began practically in 1870 and continued for ten years. 

 Of these the most conspicuous are entitled, " On the equations of 

 motion of electricity for stationary conductors " (1870); "On the theory 

 of electrodynamics" (1870-1874), and "Comparison of the laws of 

 Ampere and of Neumann for the electro-dynamic forces " (1873). Most 

 German physicists at that time deduced the laws of electro- dynamics 

 from the hypotheses of Wilhelni Weber. These were founded on the 

 laws of Newton for gravitational forces, and on Coulomb's law for 

 static electricity, according to which the intensity of the electrical 

 force transmitted with infinite velocity in all directions throughout 

 space, is directly proportional to the product of the two acting electri- 

 cal quantities and inversely as the square of the distance between 

 their points of situation. The force is repulsive when the electrical 

 charges are of the same kind, and attractive when they are of opposite 

 kinds. Weber extended the assumptions of Coulomb by introducing 

 besides the distance the velocity and acceleration with which the two 

 electrical quantities approached or receded from each other. These 

 suppositions of forces, which depend not simply on the distance but 

 also on the motion of the points of action, seem now, to be sure, to 

 contradict the results attained by Helinholtz in his earlier investiga- 

 tions, for he had showed that forces which depend on the distance and 

 velocity in general infringe the general law of the conservation of 

 energy, which holds as well for electro-dynamic as for other phenomena. 

 But he had not at that time considered the complicated case of the 

 laws of Weber, where the acceleration was introduced, and it can, 



