AN EXTENSION OF OPERATIONAL CALCULUS 341 



W'c then have, replacing (I), the system of algebraic equations: 



Sii/i + Z12J2 + • • • + Si„/„ = Fi + Gi, 

 (5) 



ZnlJl + S„2/2 + • • • + Z„nJn = Fn + G ,.. 



In these equations, G\, Gi, ■ • • , G„ denote the following summations: 

 Gi = Lnli' - QiVCnP + Lnl2" - Q2'IC,,p + • • • 



(6j 



The right hand sides of equations (5) are thus known in terms of the 

 impressed forces and the specified initial values of the currents and 

 charges. They can therefore be solved in the usual manner for 

 Ju • • • , Jn. Thus 



J. Fi ^ Gl . F2 + G2 ,Fn + Gn .-s 



Jm 7 1 7 r • ■ • i 7~ •• \' ) 



Having thus determined /i, •••,/„ as functions of p, h, • • • , In are 

 determined as functions of time by the Laplace integral equation: 



Jm{P) = r Im{t)e-'" dt, Pr > c, (8) 



Jo 



which completes the formal solution of the problem. Note that if 

 Gi — G2 = ■ ' • = G„ = 0, the solution reduces to the usual form. 



Equations (6) for Gi, •••,G„, may be written in a compact and 

 elegant form as follows: Let 



T = hllZLjJih, 



1 (9) 



Cjk 



T is then the kinetic or magnetic energy stored in the network and U 

 is the corresponding potential or electric energy. Then 



The foregoing solution is compact, elegant and formally complete. 

 In practical applications to networks of many degrees of freedom it 

 may well present formidable difficulties in computation and interpre- 



