Operational Methods in Mathematical Physics, 415 



Nevertheless, it seems to be doubtful if the ease and 

 simplicity of the operational method have been fully appre- 

 ciated ; and it is hoped that the following examples may 

 do something to call fresh attention to the merits of the 

 operational method. 



Supposing an E.M.F. applied by a battery of strength P, 

 it is usually easy to express the galvanometer current in the 

 symbolic form 



F(/0 



which is to be interpreted as 



P, 





k (0) 7 «A'(a) 



repeating the formula (4) quoted above. 



Thus, to produce a complete balance we must have 



F(0)=0, F(o) = (31) 



for all roots p = a. of A {pi) = 0. 



But, since F(p) is of lower degree * than A(p), it follows 

 from (31) that F(p) must be identically zero. Hence we 

 have the working rule f : — 



To obtain the conditions for a complete balance, put zero for 

 the galvanometer current ; this will lead to a certain algebraic 

 condition in p which must be satisfied identically, when p is 

 treated as an algebraic variable. 



It may, however, prove to be impossible to obtain a 

 complete balance : then we should make the time-integral 

 of the current zero, so that the galvanometer will show no 

 ballistic effect. 



Thus we wish to make 



icdt=0 



Jo 



I 



J |a(0) + ?**<(«/ } dt - u - 



Now, since all the roots a are negative in actual problems, 

 this leads to the two conditions 



F(0) n , v l F(«) n 



A(0; « « A'(a) 



* See § 3 below. 



t Heaviside, 'Electrical Papers,' vol. ii. p. 259. 



