510 Mr. 0. Heaviside. [Feb. 2 > 



is so general, and its execution usually presents no difficulties, whilst 

 the interpretation of the result may be valuable and instructive 



physically. 



A continued constant force of unit strength, commencing when 



t = o, may be represented by 



smnt 7 \ ,n^ 



* 



using a well-known integral. We may apply this to equation (1), if 

 desired, and obtain a particular form of solution. And from (11) we 

 see that a unit impulse is represented by 



r 

 p l = - cos nt . dn (12) 



acting at the moment t = 0. This is, of course, the basis of Fourier's 

 theorem. But, instead of the application of the fully developed 

 Fourier's theorem, it is more convenient to use (12) itself. Thus, 

 when / is an impulse acting when t = 0, we have the equation 



pfo representing the force. So, by (12), 



Ycosnt.dn (13) 



gives us a particular form of the solution arising from an impulse. 

 Take p = ni in Y to convert the quantity to be integrated to an 

 algebraical form. 



Since a continuously varying force may be represented by a suc- 

 cession of infinitesimal impulses, we see that a single time-integration 

 applied to (13), / being then a function of the time, gives us a form 

 of solution of the equation F = Y/, for any kind of / and Y that can 

 occur. It is, however, a theoretical rather than a practical form of 

 solution. For it usually happens that the definite integral is quite 

 unamenable to evaluation. The same may be often said of the solu- 

 tion (13) for an impulse, and in such cases it may be questioned 

 whether the form F = Y/ itself is not just as plain and intelligible. 

 In fact, in certain cases, a very good way to solve (or evaluate) a 

 solution in the form of a definite integral is to undo it, or convert it 

 to the symbolical form F = Y/, and then solve it by any way that 

 may be feasible. Nevertheless, it is interesting to know that we may 

 have a full solution, and the definite integrals are sometimes practic- 

 ally workable, or may be transformed to easier kinds. 



