1893.] On Operators in Physical Mathematics. 507 



Electromagnetic Operators. 



7. Perhaps the best way of beginning the subject, to obtain a 

 good idea of the nature of the operators and the advantages 

 attending their use, is through the theory of a connected system, 

 of linear electrical conductors. For the electrical equations seem, 

 to be peculiarly fitted for the illustration of abstract dynamical 

 properties in a clear manner, even when quite practical electro- 

 magnetic arrangements are concerned. We know that we may, by 

 the application of Ohm's law to every conductor (or to circuits of 

 conductors), express the steady current C in a conductor n due to an 

 impressed force e m in a conductor m by an equation 



C n = Y mn e m , (2) 



where Y mn is some algebraical function of the resistances of the con- 

 ductors usually of all the resistances, although in special cases it 

 may become independent of the values of some of them. Now, 

 suppose it is not the steady current that is wanted, but the variable 

 current when e m varies. The answer is obviously given by the same 

 equation when the function Y involves only resistances ; that is, when 

 there is no storage of electric or magnetic energy, so that Y is a con- 

 stant not involving d/dt. Then the flux and the force keep pace 

 together, and their ratio does not vary. It is, however, less obvious 

 that the same equation should persist, in a generalized form, when 

 every branch of the system is made to be any electromagnetic 

 arrangement we please which would, in the absence of its connexions 

 with the rest of the system, be a self-contained arrangement. To 

 obtain the generalized form of Y mn we have merely to substitute for 

 the resistances concerned the equivalent resistance operators. That 

 is, instead of Y = BC, where Y is voltage, C current, and B resist- 

 ance, we have an equation V = ZC in general for every conductor, 

 where Z is the resistance operator appropriate to the nature of the 

 conductor, which may be readily constructed from the electrical 

 particulars. These Z's substituted in Y mn in place of the B's make 

 equation (2) fully express the new connexion between the flux C 

 and the force e m . There is much advantage in working with resis- 

 tance operators because they combine and are manipulated like 

 simple resistances. Of course (2) is really a differential equation, 

 though not in the form usually given. To make it an ordinary 

 differential equation we should clear of fractions, by performing such 

 operations upon both sides of (2) as shall remove denominators and 

 all inverse operations. It is then spread out horizontally to a great 

 length (usually) and becomes very unmanageable. Also, we lose sight 

 of the essential structure of the operator Y. 



8. By arrangements of coils and condensers in our linear system 



