1893.] On Operators in Physical Mathematics. 



difficulty. Suppose it is zero before and constant after the moment 

 t = 0. Then we shall have 



A combination of direct and inverse operations, which frequently 

 occurs in the theory of waves, is exemplified in 



F = <rprl'(a + bp~ l + cp~ 2 + . . . . )/. (G) 



Here we may perform the integrations first, getting the result 



0() say, and then let the exponential operate, giving, by Taylor's 

 theorem, 



F=0(*-r/v). (7) 



Or we may let the exponential operator work first, and then perform 

 the integrations. This may be less easy to manage, on account of 

 the changed limits. 



Two important fundamental cases, which constitute working 

 formulae, are 



v =~f' and F =^> < 8 > 



with unit operand, that is, /=0 before and constant after = 0. 

 Here we may expand in inverse powers of p, getting, in the first 

 case 



F = (i + ap-> + a?p-*+. . ..).= e<, (9) 



and in the second case c~ at . The latter expresses the effect of a 

 unit impulse in a system having one degree of freedom, with friction, 

 as when an impulsive voltage acts upon a coil. 



Solutions for Simple Harmonic, Impulsive, and Continued Forces. 



10. A very important case, admitting of simple treatment, occurs 

 when the force is simple periodic, or a sinusoidal function of the 

 time. It may happen that the resulting state of F is also sinus- 

 oidal. For this to occur, there must be dissipation of energy, to 

 allow the initial departure from the simple periodic state to subside. 

 We then have^ 2 = n* applied to F as well as /, where w/2?r is the 

 frequency ; so that the substitution of ni for p in Y brings equation 

 (1) to the form 



F = (Y +Y lt -)/ = (Y + Y^-/, (10) 



where Y and Y! are functions of rc 2 . We now find F by a simple 

 direct operation. This case is so important because its application 



