SCIENCE. 



[N. S. Vol. VIII. No. 206. 



ductions, and it was then found that their 

 quality was also exceptional. His writings 

 have mainly dealt with a quite different 

 sort of subject from those enumerated above, 

 and have treated either the flow of variable 

 currents in wires or the transmission of 

 electro-magnetic waves in free space. As 

 early as 1876, in a paper modestly entitled 

 'On the Extra-current,' he gives for the 

 first time the partial differential equations 

 for the propagation of current and potential 

 along wires, and treats them by the methods 

 of Fourier. Later he considers the most 

 complicated questions caused by the ter- 

 minal conditions involved by the introduc- 

 tion of various sorts .of electrical apparatus, 

 such as those used in telegraphy and te- 

 lephony. These papers will well repay the 

 attention of pure mathematicians, to whom 

 they offer a host of questions for rigid treat- 

 ment. For instance, to fix the ideas, we 

 have the equation of propagation 



+ 6- 



with the condition that for two or more 

 given values of x, f is to satisfy given linear 

 ordinary differential equations in t, and 



thatfor a given value of <, <p and -^^ are to be 



at 



given functions of x. 



If we attempt to satisfy the equation by 

 particular solutions which are trigonometric 

 functions of the time we get an ordinary 

 differential equation in x, and if we then 

 make use of trigonometric functions of mul- 

 tiples of X the multiples allowable will be 

 determined by certain transcendental equa- 

 tions according to the terminal conditions. 

 The function <p is then to be developed in a 

 series of such trigonometric terms. The 

 nature of the proofs desired relating to the 

 series may be readily inferred. The papers 

 by Heaviside are extremely numerous and 

 bulky, and it is desirable that the methods 

 there used should receive critical attention 



from mathematicians, for it must be said 

 that Heaviside uniformly disdains such 

 things as existence-theorems, depending 

 chiefly on bis intuitions drawn from phys- 

 ical reasoning. 



A large portion of Heaviside's labors has 

 been devoted to the systematization and 

 extension of Maxwell's theory and the 

 attempt to disseminate a knowledge of that 

 theory among the physical public. His 

 results agree so nearly with those of Hertz 

 that I shall give them in the notation and 

 form used by the latter, which seem to me 

 preferable. The attempt to bring out the 

 symmetry or reciprocity between electrical 

 and magnetic phenomena has been para- 

 mount with both Heaviside and Hertz. 

 Accordingly, we have for the connections 

 between the electric and magnetic field in- 

 tensities, represented respectively by X, Y, 

 Z and L, M, N, and the corresponding in- 

 ductions or polarizations f , |, % and IT, P^, ^, 



(11) M = ^r- 



St = /«il!f (12) 

 |t = //iV 



instead of equations (2) and (5), e repre- 

 senting the specific inductive capacity and 

 sj. the magnetic permeability. For the elec- 

 trical and magnetic densities i\ and />„ we 

 have 



(13) 

 (14) 



- + 

 3a; ^ 



31 3i 

 33/ "^ 32. 



47rl 



3f 3| 

 3a; "•" 3i 



■ + 



For the mutual connections of the two 

 fields we have 



35 _3JV__3iU' 



4 TTM -[- 



3<' 



(15) 



471-11 + — ==- 



dt 



3f 

 ■dt '' 



92 



3jr_ 



' 3x 



^ dZ 

 "33/ " 



32 



3iV 

 3x ' 



3L 



"32/' 



dV 



dz' 



