PRACTICAL APPLICATION OF THE FOURIER INTEGRAL 657 



At the limit a = 0, the cause and effect would be the singular ^„ 

 or #_„ functions. 



The curves on the right for w = of Fig. 3 and w = 1 of Fig. 4 show 

 that at the limit a= a unit step in the voltage produces a unit impulse 

 in the current through a unit condenser; on the other hand, a unit im- 

 pulse applied to a unit inductance gives a current which is a unit step. 



The curves of Fig. 2 may be used to furnish another illustration of 

 the use of coefficient pairs, in connection with the problem of finding 

 networks in which assigned transient currents will be produced by 

 assigned impressed electromotive forces. Any curve n being the 

 assumed cause and the next curve (w + 1) the assumed effect, the 

 required admittance is 0n+i(/)/[«0n(/)]- This admittance is pre- 

 sented by a ladder network of (n -{- 1) elements: perfect inductance 

 coils in the series arms, perfect condensers in the shunt arms, the ladder 

 starting with a shunt condenser, the values of the shunt capacities 

 being equal to 2, 2n(n — 1)~S 2n{n — l)~^(w — 2)(n — 3)~S etc., 

 and the values of the series inductances being equal to (27r»)~\ 

 {2irn)~^(n — l){n — 2)~\ etc. In verifying the solution of this prob- 

 lem, it is to be noticed that the mates of the curves n and (w + 1), 

 regarded as impulse coefficients, are the same curves multiplied by i~" 

 and ^~("+i); the quotient of the latter mate divided by the former 

 mate is the admittance of the network as given above. 



On the other hand, any curve (n + 1) being the cause, the curve n 

 is the effect in the reciprocally related ladder network of (n + 1) 

 elements, starting with a series reactance coil, the values of the series 

 inductances being equal to 2, 2n(n — 1)"^ 2n(n — l)~^{n — 2){n — 3)~S 

 etc., and the values of the shunt capacities being equal to (27rw)~S 

 (27rw)-Kw - \){n - 2)-i, etc. 



Practical Applications of Coefficient Pairs in Table II. 



In general, each of the three subsidiary problems employed by 

 Fourier is unsolvable in closed form. In a strictly limited number of 

 cases, however, all three problems have been solved and the final 

 transient solution obtained. These solutions should be cherished and 

 collected for ready reference. It is a needless waste of time to repeat 

 the analytical work each time a solution is required. Except for a 

 few special cases lying outside of the scope of the table, all practical 

 applications of closed form coefficient pairs which were found in a 

 preliminary search are included in the transient solutions of Table II. 

 As it stands, the table is far from a complete list of closed form solu- 

 tions, but it contains many important solutions and serves to illustrate 

 the use of Table I. Table II contains 39 admittances, with references 

 to 39 systems which serve to illustrate the occurrence of these admit- 



