1893.] On Operators in Physical Mathematics. 513 



R and L are themselves differentiating operators of complicated form 

 in general, or, more strictly, R + Lp is a resistance operator, say R", 

 of very complex form. But it is quite sufficient to take the form 

 R + Lj9, where R is the effective resistance and L the inductance per 

 unit length of circuit. S and K mean the permittance and leakage 

 conductance per unit length. 



Now we may readily obtain the simple periodic solution out of (17), 

 by the before-mentioned substitution p = ni ; and in doing so we may 

 use the general operator R", for that will then assume the form 

 R-f-Lp- From this solution a wholly uninterpretable definite integral 

 can be derived to express the effect of an impulse or of a steady im- 

 pressed force. The question was, how to obtain a plain understand- 

 able solution from (17) itself to show the effect of a steady force. 

 To illustrate, we may here take merely the case in which K = 0, whilst 

 R and L are constants, because the inclusion of K (to be done later) 

 considerably complicates the results. We have then to solve 



where a is a constant and p = djdt. The operand is understood to 

 be unity, that is, / = before and =1 after t = 0. It is needless to 

 write unit operands, and it facilitates the working to omit them. 

 Now, the first obvious suggestion is to employ the binomial theorem 

 to expand the operator. This may be done either in rising or in 

 descending powers of p. Try first descending powers, since by ex- 

 perience with rational operators we know that that way works. We 

 have 



The integrations, being separated from one another, can be imme- 

 diately carried out through p~ n = t n /\n, giving the result 



at . I . 3 /at\* 1.3. 



or, which is the same, 



' (21) 



wliere I is the well-known cylinder function, Now, that this re- 

 sult is correct may be tested independently, viz., by its correctly 

 satisfying the differential equation concerned and the imposed con- 

 ditions. We therefore obtain some confidence in the validity of the 

 process employed. 



