December 23, 1904.] 



SCIENCE. 



865 



The fundamental problems of static elec- 

 ti'icity belong- to the same order of ideas, 

 and that was surely a beautiful triumph 

 for theory, the discovery of the celebrated 

 theorem on electric phenomena in the in- 

 terior of hollow conductors, which later 

 Faraday rediscovered experimentally, with- 

 out having known of Green 's memoir. 



All this magnificent ensemble has re- 

 mained the type of the theories already old 

 of mathematical physics, which seem to us 

 almost to have attained perfection, and 

 which exercise still so happy an influence 

 on the progress of pure analysis in suggest- 

 ing to it the most beautiful problems. The 

 theory of functions offers us another mem- 

 orable affiliation. 



There the analytic transformations which 

 come into play are not distinct from those 

 we have met in the permanent movement 

 of heat. Certain fundamental problems 

 of the theory of functions of a complex 

 variable lost then their abstract enuncia- 

 tion to take a physical form, such as that 

 of the distribution of temperatiire on a 

 closed surface of any connection and not 

 radiating, in calorific equilibrium with two 

 sources of heat which necessarily corre- 

 spond to flows equal and of contrary signs. 

 Transposing, we face a question relative 

 to Abelian integrals of the third species 

 in the theory of algebraic curves. 



The examples which precede, where we 

 have envisaged only the equations of heat 

 and of attraction, show that the influence 

 of physical theories has not been exercised 

 only on the general nature of the problems 

 to be solved, bi^t even in the details of the 

 analytic transformations. Thus is cur- 

 rently designated in recent memoirs on 

 partial differential equations under the 

 name of Green's formula, a formula in- 

 spired by the primitive formula of the 

 English physicist. The theory- of dynamic 

 electricity and that of magnetism, with 

 Ampere and Gauss, have been the origin of 



important progress; the study of curvi- 

 linear integrals and that of the integrals 

 of surfaces have taken thence all their de- 

 velopments, and formulas, such as that of 

 Stokes which might also be called Ampere 's 

 formula, have appeared for the first time 

 in memoirs on physics. The equations of 

 the propagation of electricity, to which are 

 attached the names of Ohm and Kirchoff, 

 while presenting a great analogy with ti\ose 

 of heat, offer often conditions at the limits 

 a little different; we know all that teleg- 

 raphy by cables owes to the profound dis- 

 cussion of a Fourier's equation carried 

 over into electricity. 



The equations long ago written of hydro- 

 dynamics,, the equations of the theory of 

 electricity, those of Maxwell and of Hertz 

 in electromagnetism have offered problems 

 analogous to those recalled above, but un- 

 der conditions still more varied. Many 

 unsurmounted difficulties are there met 

 with; but how many beautiful results we 

 owe to the study of particular cases, whose 

 number one would wish to see increase. 

 To be noted also as interesting at once to 

 analysis and physics are the profound dif- 

 ferences which the propagation may pre- 

 sent according to the phenomena studied. 

 With equations such as those of sound, we 

 have propagation by waves ; with the equa- 

 tion of heat, each variation is felt instantly 

 at every distance, but very little at a very 

 great distance, and we can not then speak 

 of velocity of propagation. 



In other cases of which Kirchoff 's equa- 

 tion relative to the propagation of elec- 

 tricity with induction and capacity offers 

 the simplest type, there is a wave front 

 with a velocity determined but with a re- 

 mainder behind which does not vanish. 



These diverse circumstances reveal very 

 different properties of integrals; their 

 study has been delved into only in a few 

 particular cases, and it raises questicns 



