900 



SCIENCE 



[N. S. Vol. XXXVIII. No. 991 



matieal physics treated by Poineare was 

 that of electrical waves and oscillations. 

 The reason for this is not far to seek. Not 

 only are the equations of Maxwell's theory, 

 complicated though they seem to be and 

 involving a large number of vectors, ca- 

 pable of reduction to one of the forms al- 

 ready mentioned, but their solution calls 

 for great knowledge of differential equa- 

 tions and even in the last few years in the 

 hands of Poineare permits of being treated 

 by integral equations. Besides this the ap- 

 plications to the subject of wireless teleg- 

 raphy are of great practical importance as 

 well as theoretical interest. If we write 

 Maxwell 's equations in the simplest ease, 

 _SH_56 

 ""By Sz' 



' ' ' ^ ' ' ' ' Sf .Ss ^Sk 



■' dt 6n' 



Sa ,SpSy ^Q 



, I , Sf\ Sy S0 dx Sy Sz ' 



where («, /8, y) denotes the magnetic field, 

 47r (/, g, h,) the electric field, u, v, iv the 

 conduction current, p the electric density, 

 * the electric scalar potential, {F, G, H) 

 the vector-potential, where the vector-po- 

 tential is defined by the differential equa- 

 tions 



^sH _sG 



" " Sy dz' 



there is still something indeterminate in 

 this vector. Maxwell assumes 



8x Sy'^ Sz ' 



which does not lend itself to simplicity, but 

 if instead we put, as was later done by 

 Lorentz, 



Sx Sy '^ Sz St ' 

 we get a great simplification. It is notable 

 that this was done by Poineare in 1893 in 

 his lectures on electrical oscillations, evi- 



dently quite independently of Lorentz. 

 Our equation for the vector-potential then 

 reduces to the form 



—pr — Af = 4jrM, 



St' 



which shows that if there is no current 

 anywhere the vector-potential is propagated 

 in waves. The scalar potential also satis- 

 fies the equation 



S'<P 



^ - A^ = 4.p. 



It was shown by Lorentz in 1892 and inde- 

 pendently in the same year by Beltrami, 

 that this equation, which, in case the first 

 term vanishes, reduces to that of Poisson, 

 is satisfied by a potential function 



in which, however, p' represents the value 

 of p not at the instant in question but at an 

 instant precedent by the time required to 

 come from the point of integration with 

 the wave-velocity. Such a potential is now 

 known as a retarded potential and I may 

 be allowed for a moment to digress upon 

 the interesting history of this Lorentz-Bel- 

 trami equation. Its properties with those 

 of the retarded potential are given by 

 Poineare in his lectures in 1893, evidently 

 quite independently of Lorentz. It turns 

 out, however, that the properties of the 

 equation are given in Lord Rayleigh's 

 "Theory of Sound," appearing in 1877. 

 This, however, is not its first appearance, 

 for we find the same equation in a paper by 

 L. Lorenz, of Copenhagen, in 1869, on elas- 

 ticity. More remarkable yet in a paper 

 presented by Riemann to the Eoyal Society 

 of Gottingen in 1858, but published only in 

 1867 after Riemann 's death, we find the 

 same identical equation for the electric po- 

 tential although deduced from considera- 

 tions which are now untenable. It is curi- 

 ous to remark that although this equation is 

 mentioned by Maxwell in the last chapter 



