508 Mr. 0. Heaviside. [Feb. 2 r 



we may construct an infinite variety of resistance operators, and of 

 conductance operators, such as Y mn above. They are, however, always 

 algebraical functions of jt?, and are finite. If expanded, equation (2) 

 always becomes an ordinary linear equation of a finite number of 

 terms. But if we allow conduction in masses, or dielectric displace- 

 ment in masses (with allowance for propagation in time), the finite 

 series we were previously concerned with become infinite series. 

 This, at first, appears a complication, but it may be quite the reverse, 

 for an infinite series following an easily recognized law may be more 

 manageable than a finite series. Still, however, the equivalence to 

 ordinary differential equations persists, provided our arrangement is 

 bounded. But when we remove this restriction, and permit free dis- 

 sipation of energy in space (or equivalently), another kind of operators 

 comes into view. The complexity of the previous, due to the reaction 

 of the boundaries, is removed; simpler forms of operators result, and 

 they do not necessarily admit of the equations taking the form of 

 ordinary differential equations, as they may be of an irrational nature. 

 This brings us necessarily to the study of generalized differentiation y 

 concerning which, more presently. 



Operators admitting of Easy Treatment. 



9. In the meantime, notice briefly some of the ideas and devices 

 that occur generally in the treatment of operators. First of all, we 

 may obtain the steady state of F due to steady/, when there can be 

 a steady state of F, by simply putting p = in the operator Y con- 

 necting them, p meaning d/dt. Even when there is no resultant 

 steady state of the flux, as when reflections from a boundary continue 

 for ever, the term F = Y / has its proper place and significance. 



Next, we may notice that if the form of Y should involve nothing 

 more than separate differentiations, as in 



....)/, (3) 



then all we have to do is to execute the differentiations to obtain F 

 From /. When / is a continuous function, this presents nothing 

 special. When discontinuous, however, a special treatment may be 

 needed. 



In a similar manner, there may be only separate integrations or 

 inverse differentiations indicated in Y, as when 



+ ....)/. (4) 



Since/is a definite function of the time, so are its successive time- 

 itegrals. In this case,/ may be discontinuous, and yefc present no 



