44 BELL SYSTEM TECHNICAL JOURNAL 



cuius. This formula is presented and discussed in the present paper 

 with the hope that it may be of service to students of Heaviside in 

 understanding and applying his methods. The paper may also serve 

 as a brief recapitulation of some of the outstanding methods ap- 

 plicable to the solution of problems in electric circuit theory. 



The problems to which Heaviside applied his operational calculus 

 relate to the oscillations of electrical and mechanical systems which 

 can be described by a system of linear differential equations with 

 constant coefficients or to such partial differential equations as the 

 telegraph equation. The system is supposed to be set into oscillation 

 from a state of equilibrium by suddenly applied forces and the opera- 

 tional formula gives the resulting behavior of the system. 



The type of problem to which the operational calculus is applicable 

 and Heaviside's method of solution may be illustrated in a sufficiently 

 general manner for didactic purposes in connection with the solution 

 of the system of equations 



aiixi -\- . . . -\- ai„x„ = Fi(t) 

 (1) 



a„ixi -f . . . + a„„x„ = F„(t) 



where the coefficients ajk are in general polynominals of the form 



ajk = ajk -f- Pjk ^ + '^^'^ ^ + • • • 



and the a, /3, 7 coefficients are constants. 



We are concerned with the determination of the variables Xi . . x„ 

 as functions of the independent variable t for the following boundary 

 conditions; the known functions F] . . F„ and the variables Xi . . :v:„ 

 are identically zero for values of the independent variable t^O. In other 

 words, the system is initially in a state of equilibrium when the 

 " forces " Fi . . . F„ are applied. These boundary conditions are 

 extremely important in physical problems. 



Owing to the linear character of the equations we may without loss 

 of generality set all the F functions equal to zero except one, say 

 Fi{t), and write 



anXi + . . . + ai„x„ = F(t) 



(3) 



a„iXi -}-...+ a„„x„ = 0. 



The solution of equations (3) for the prescribed boundary condi- 

 tions may be made to depend on the auxiliary equations in the auxil- 



