I'h-iii.wii'oixi iMri.n.ixci: ()/• nio-Misii ciHcriis 657 



luuu fretiia-ncit's ;uul associalnl (l,ini|>iii>; nm^taiits) of tlu' circuit 

 from the known values of the elements. This prol)lem is intimately 

 relateii to the deterniination of the <lri\inj;-p"''" im|)e<iance of the 

 circuit, since the free ix-riods of the circuit can he found by setting 

 the drivinn-|X)int im|H-dance in any one mesh equal to zero.* By 

 this method the free iH-ri(Hls are found as the r<H)ts of an equation of 

 the fourth denree,^ the exact solution of which involves, in general, 

 cuml)ersome formulas. In order to ohtain formulas which are better 

 adapted to numerical computation, \arious approximations are 

 usually made.'' 



This electrical problem of the free o>ciliati<ins of a circuit is formally 

 the same as the dynamical problem of the small oscillations of a 

 system alxiiit a position of ecjuilibrium. The determination of the 

 free [x*ri<Kls of a circuit can be made directK' from the solution of this 

 d\namical problem." 



The first part of this pa|)er treats a much more general problem 

 than the determination of the driving-point impedance of a particular 

 circuit from the given values of the elements, nameK', the determina- 

 tion of the entire range of possibilities, together with the inherent 

 limitations, of such an impedance. The method employed is to find 

 the general form of the impedance as a function of the time coefficient, 

 .md then to investigate the restrictions which must be satisfied by a 

 function of this character in order that it may represent an impedance 

 realizable by means of a circuit consisting of resistances, capacities, 

 and inductances. In the present paper, this in\estigation is limited 

 to the dri\ing-point inifK'dance of a two-mesh circuit; the driving- 

 }K>int impedance of an «-mesh circuit will be treated in a future paper. 



The driving-point imiiedance of any circuit containing no resistances 

 has been investigated in a previous paper,"' where it has been shown 

 that any such impedance is a pure reactance with a number of resonant 

 and anti-resonant frecjuencies which alternate with each other, and 



•C. A. Campbell, Transactions of the .1. /. E. B.. ii), mil, p.igcs 87,?-9(W. 



'.An exhaustive discussion of this fourth degree e<|uation has Ix-en given liy J. 

 Soninier, Annalrn der Physik, fourth series, 58, l')I9, pages .?75-.?')2. 



'■ For tvpiral niethcxls of solution see the [xipers of I.. Cohen, BulUlin of the Burrau 

 of Standards, 5, 1908-9, pages 511-541; B. .Marku, Jahrhuch der drahtlosen Tele- 

 graphie und Telephonie, 2, 1909, p;iges 251-293; \'. Bush, Proceedings of the I. R. E., 

 5, 1917. ixiges .?6.1-382. 



' Representative investigations of this dynainital problem are those of Lord 

 Rasleigh, Proceedin/^s of the London Mathematical Society, 4, 1873, pages 357-368, 

 Philosophical Magazine, fifth scries. 21, 1886, pages 369-381, anil sixth series, 3, 

 l'«)2. (Mges 97-117 ("Stientitic Papers." I, 17(> 181, II. 475-485, and V, 8-261; 

 K. J. Routh, ".Advanced Rigid Dynamics," sixth edition, 1905, pages 232 243; 

 .A. <".. Webster. "Dynamics." second edition, 1912, f)ages 157-164. 



'K. M l..-.>.r. H.-ll sv,7,.,„ fWhnicat Journal, 3, 1924, (xiges 259-267. 



