January 25, 1895.] 



SCIENCE. 



105 



who measured the wave length (about 3 

 metres in the earliest experiments ) of the 

 waves produced by these rapid oseillations 

 by means of the intensity of the spark in 

 the spark-gap of a secondary circuit, the 

 so-called resonator. Tlie period was ciileu- 

 lated by the Thomson formula and divitling 

 the wave-length by the period gave the ve- 

 locity of propagation, which, according to 

 the Faraday-JIaxwell theory, should be 

 equal to that of light, and that, too, both in 

 the immediate vicinity of the conductors 

 and in the dielectric. A mere sketch of 

 these experiments is given for the purpose 

 of outlining the plan of the discussion to be 

 carried out in the succeeding chapters of 

 the book. Hertz's method of calculating 

 the period of his oscillators is reproduced 

 more or less faithful]}- and the various ob- 

 jections against it discussed. 



Theory of Hertzian O.iciUations. — This dis- 

 cussion paves the way gradually for the gen- 

 eral theory of the Hertzian oscillator to be 

 taken up in the next chapter. This theory 

 can be described as the mathematical dis- 

 cussion of the following problem : Given a 

 homogeneous dielectric extending indefi- 

 nitely. This dielectric is acted upon by a 

 steady electrical force applied at a conduc- 

 ductor, the oscillator. It is therefore elec- 

 trically strained. Describe the pi-ocess by 

 means of which the dielectric returns to its 

 neutral state when the mitial electrical 

 strain is suddenly released. 



The discussion uuist necessarily start from 

 Maxwell's fundamental equations. They 

 are in the form given bj' Hertz, partial 

 differential equations connecting the com- 

 ponents of tlie electric and of the magnetic 

 forces at any point in tlie dielectric. Hence, 

 using the language of the mathematician, 

 the solution of the above problem will 

 consist in the integration of Maxwell's dif- 

 ferential equations, which, translated into 

 the language of the experimental physicist. 

 means that the solution will cimsist in Hud- 



ing the resulting electrical wave, that is, 

 its period, its decrement due to radiation 

 and dissipation, and its direction and 

 velocity of propagation. It is evident, 

 therefore, both to the mathematician and 

 to the physicist that the conditions at tlie 

 boundary surfaces separating the dielectric 

 from the conductor must first be settled. 

 To these Poincar6 devotes careful attention. 

 A lucid demonstration is given of the theo- 

 rems that in the case of rapid oscillations 

 there will be : a. Very slight penetration of 

 the current into the conductor; b. A 

 vanishing of the electric and the magnetic 

 force in the interior of the conductor, c. 

 Electric force normal and magnetic force 

 tangential to the surface of the conductor, 

 etc. 



Then follows a beautiful mathematical 

 solution of the general problem mentioned 

 above. It is this : The law of distribution 

 of the conduction current on the oscillator 

 being given the electric and magnetic force, 

 and tlierefore the state of the wave, at any 

 point in the dielectric and at any moment 

 can be calculated by a simple differentiation 

 of a quantity called the vector potential. 

 This quantity is determined from the cur- 

 rent distribution in a manner which is the 

 same as that employed in the calculation of 

 the electrostatic potential from the disti-ibu- 

 tion of the electrical charge, but on the sup- 

 position that the force between the various 

 points of the dielectric and the surface of 

 the oscillator is propagated with the velocity 

 of light. The value of this solution rests 

 on tlu! fact that the law of distribution of 

 the conduction current can be closelj' esti- 

 mated in some oscillators, as, for instance, 

 in the case of Bloiidlot's oscillator consisting 

 of a wire bent so as to form a rectangle in 

 one of whose sides a small plate condenser 

 is interposed. A special form of this vector 

 potential applicat)le to oscillatoi-s whose 

 surfiice is tiiat of revolution is deduced and 

 applied to Lodge's spherical oscillator. 



