506 Mr. 0. Heaviside. [Feb. 2, 



We need not assert that the determinateness of F from equation (1) 

 is true for all forms of the function Y that may be written down 

 arbitrarily ; but that it is true in the forms presenting themselves in 

 dynamical problems seems to be necessitated. 



5. We have, therefore, presented to us the problem of solving this 

 equation for any particular form of Y that occurs. This may be 

 very easy and obvious, or it may be excessively difficult and obscure. 

 In the latter case it may be so merely because we do not know how 

 to do it. Then we should find out. As our argument is that Y finds 

 F from /definitely, there should be definite rules for the manipula- 

 tion of the operator Y, or of the expression Y/, for its conversion to 

 the form of an ordinary mathematical function, which will be the 

 solution in the usual sense, freed from differentiating operations. We 

 may find how to work by experiment. For, if two different methods 

 lead to different results, one of which we find to be correct by in- 

 dependent tests, we can safely assert that one of the methods was 

 partly wrong, whilst the other may have been wholly correct. So by 

 practice we may come to know something about it. 



6. Again, the function Y, regarded as an algebraical function, may 

 admit of different forms of expression. These are algebraically 

 equivalent, but to what extent they may be equivalent in their ana- 

 lytical aspects for instance, one series involving differentiations 

 equivalent to another involving integrations, and leading to results 

 which are either identical or equivalent cannot be safely said 

 beforehand. It is, in its generality, a rather difficult and obscure 

 matter. In special cases I find that forms of Y which are algebrai- 

 cally equivalent are also analytically equivalent ; but I have not 

 succeeded in determining the amount of latitude that is permissible 

 in the purely algebraical treatment of operators. JSTo doubt there are 

 definite limitations, but they have to be found. I have, however, 

 extensively employed the algebraical treatment experimentally,* sub- 

 ject to independent tests for guidance. It proved itself to be a 

 powerful (if somewhat uncertain) kind of mathematical machinery. 

 We may, for example, do in a line or two, work whose verification 

 by ordinary methods may be very lengthy. On the other hand, the 

 very reverse may be the case. I have, however, convinced myself 

 that the subject is one that deserves to be thoroughly examined and 

 elaborated by mathematicians, so that the method may be brought 

 into general use in mathematical physics, not to supplant ordinary 

 methods, but to supplement them ; in short, to be used when it is 

 found to be useful. As regards the theory of the subject, it is in- 

 teresting m an unusual degree, and the interest is heightened bv the 

 mystery that envelops certain parts of it. 



my ' Electrical Pa P e < l. 2, of the treat- 

 well as rational operators. 



