1891.] Application to Periodic Electric Currents. 



205 



Since the parts of o 2 may be either positive or negative, there is 

 nothing to hinder its evanescence by compensation. In the above 

 combination of an electromagnet and condenser compensation occurs 

 when p*LC = 1, that is, when the natural period with terminals 

 connected coincides with the forced period. The combination is 

 then equivalent to a simple resistance ;* but a variation of fre- 

 quency will give rise to a positive or negative 0%. 



The case of two electromagnets in parallel is treated in my paper 

 on w Forced Harmonic Oscillations ;"f and other combinations have 

 been discussed by Mr. Heaviside and myself. But the above examples 

 will suffice to illustrate the principle that the relation of V to x is 

 one of proportionality, and may be expressed by the single complex 

 symbol a. We fall back at any time upon the case of mere resistance 

 by supposing a to be real. In like manner 6, c, d, e, and /are sym- 

 bols expressing the electrical properties of the remaining branches. 



In all electrical problems the generalised quantities a, 6, &c., com- 

 bine, just as they do when they represent simple resistances. Thus, 

 if a, a be two complex quantities representing two conductors in 

 series, the corresponding quantity for the combination is a-j-a'. 

 Again, if a, a' represent two conductors in parallel, the reciprocal of 

 the resultant is given by addition of the reciprocals of a, a'. For, if 

 the currents be x and x, corresponding to a difference of potentials 

 V at the common terminals, 



V = ax ax, 

 x+x = 



so that 



The investigation of the currents in networks of conductors is 

 usually treated by " KirchhofPs rules," and this procedure may of 

 course be adopted in the present case to determine the current 

 through the bridge of a Wheatstone combination. But it will be 

 more instructive to put the argument in the form applicable to the 

 forced vibrations of all mechanical systems which oscillate about a 

 snfiguration of equilibrium. 



If p/27r represent the frequency of the vibration, the coordinates 

 fa, fa, fa' determining the condition of the system, and the cor- 

 Jsponding forces i^, i^, ^r 3 . . . . are all proportional to e'*', and the 

 jrdinates are linear functions of the forces. J For the present 

 Jurpose we suppose that all the forces vanish, except the first and 

 3ond. Thus fa, fa are linear functions of l ir l and ^r z , and, con- 

 rersely, * b i^ may be regarded as linear functions of fa and fa. We 

 ly therefore set 



* ' Theory of Sound,' 46, MacmiUan, 1877. 

 t ' Phil. Mag.,' May, 1886. 



* ' Theory of Sound,' vol. 1, 107. 



