1893.] On Operators in Physical Mathematics. 505 



matical simplicity, we may have a space-distribution of force or of 

 displacement given, whose effect is required. 



To answer the question, we may investigate the general differential 

 equation of the system, find its solution (series, integrals, &c.), and 

 then introduce special values of constants or of functions to limit the 

 generality of the problem, and bring the solution to satisfy the re- 

 quired conditions. Details may differ according to circumstances, 

 but this may serve to describe the usual process. 



2. There is, however, a somewhat different way of regarding the 

 question. We may say that we have no special concern with the 

 general solution which would express the disturbance anywhere due 

 to initial energy throughout the system ; but that we have simply a 

 connected system, a given point (for example) of which is subjected 

 to impressed force, communicating energy to the system, and we 

 only want to know the effects due to this force itself. Since, 

 therefore, the connexions are definite, we must have some definite 

 connexion between the "flux " F and the " force "/, say 



F=Yf, (1) 



where Y is a differentiating operator of some kind, a function of d/dt, 

 the time- differentiator, for instance, when the connexions are of a 

 linear nature. Here / is some given function of the time, and Y 

 indicates the performance upon / of certain operations, whose result 

 should be to produce the required function F. 



3. An important point to be noted here is that there is, or should 

 be, no indefiniteness about the above equation. The operator Y 

 should be so determined as to fully eliminate all indeterminateness, 

 and so that the equation contains in itself the full expression of the 

 connexion between the force and the flux, without any auxiliary con- 

 ditions, or subsequent limitations, except what may be implicitly 

 involved in the equation itself. 



Determinateness of a Solution through the Operator. 



4. But as soon as we come to distinctly recognize this determin- 

 ateness of connexion, another point of important significance presents 

 itself. It should be possible to find F completely from/ through the 

 operator Y without ambiguity and without external assistance. 

 That is to say, an equation of the form (1) not only expresses a 

 problem, but also its solution. It may, indeed, not be immediately 

 interpretable, but require conversion to some other form before its 

 resultant meaning can be seen. But it is, for all that, a particular 

 form of the solution, usually a condensed form, though sometimes it 

 may be of far greater complexity than the full ordinary solution. In 

 this respect the nature of the function / is of controlling importance. 



