124 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



which are either conducting or refracting with respect to a vacuum, or 

 have values of the dielectric and magnetic constants different from 

 those of a vacuum. From the motions of the ponderable parts the 

 corresponding motions of the ether may be determined. For space 

 free from ponderable masses and filled only with ether, the question 

 arises, if pure ether is free from inertia whether it makes way for the 

 motion of material bodies through it, or penetrates them, remaining in 

 either case wholly at rest, or whether it in part moves with them and 

 in part makes way for their passage. Under the assumption that the 

 ether has, mechanically considered, the properties of a Motionless, 

 incompressible liquid, but without inertia, Helmholtz showed that the 

 laws founded by Maxwell and elaborated by Hertz are sufficient to 

 completely determine the laws of the changes and motions which take 

 place in the ether. The recapitulation of the laws of electrodynamics 

 under the principle of least action, as already made by Helmholtz, 

 required for this purpose oidy the introduction of the hypothesis of 

 uncompressibility. This would be accomplished by supplying in the 

 expression for electrokinetic potential the left-hand member of the 

 definition equation of incompressibility. In this way important con- 

 clusions could be drawn concerning the rise and decline of pondero- 

 motive forces in ether at rest and in motion. 



In an incomplete " Supplement to the paper : On the principle of least 

 action in electrodynamics" (1894), Helmholtz returned once more to 

 his recapitulation of the Maxwell-Hertz laws of electrodynamics in the 

 generalized form of the principle of least action, in order to decide 

 whether the observed values of the total energy of electromagnetic 

 processes required the addition of a linear function of the velocities, 

 and where this is the case to derive the expression for the.ponderomo- 

 tive forces from this principle. 



Here ended the long series of brilliant mathematical and mathemati- 

 cal-physical researches of this incomparable investigator, which, so far 

 as was possible without going deeply into the refinements of his mathe- 

 matical analyses I have endeavored to outline before you, although 

 certainly in an incomplete manner. What might have been the con- 

 tents of the address which Helmholtz was preparing for the quadren- 

 nial scientific assemblage at Vienna with the title, " On continuous 

 forms of motion and apparent substances," and of which he left but a 

 few pages of manuscript for the introduction, will forever remain 

 unknown to us. But one may well surmise that the world of science 

 would have found there the philosophic kernel of the great researches 

 which he carried out in the last years of his life on the foundations and 

 principles of mechanics and physics. 



