6&2 



BELL SVSTEM TLCIIMCAL JOURNAL 



These coefficients Ljk. Rjk, and C'* are siiljject to the ener^\' con- 

 ditions stated abo\'e, namcK'. lliai ihe magnetic energx', the dissipa- 

 tion, and the electric enerj:\-, 



-oH ^^"'j'"- 2 2 ^*'-''*' '"^^ -9 ]S 2 7^ ( '>^' ('>-■<"' (39) 

 ^j = \k^] 3=1 k=i ^ j=ik^\ '-■''' ' 



respectively, are each positive for any possible distribution of the 

 currents {ij, ik, ...)'" t^he branches of the circuits.'- In other words, 

 the coefficients Ljk, Rjk, and 1 Q* are subject to the condition that 

 the three quadratic forms of which these are the coefficients must be 

 positive for all real \'alues of the \ariables. All the iirinci|)al minors 

 of the determinants 



Lii L12 . . . Lin 

 L21 1-22 ... Z,2„ 



/.„. L„ 



. In 



Rii Rii . . 

 i?2i R22 . . 



Rni R,r2 . . 



and 





'Cn„ 



(40) 



are positixe or zero In- \irtue of this condition. '^ This same condi- 

 tion holds for the inductances if the coefficients Ljk apply to branches 

 instead of meshes. 



By expanding the determinants in the numerator and denominator 

 of the expression for the driving-point impedance given above, we find 



A _ao{ ip)'+a,iip)'- ^ + ai{ip)"-^+. . .+a2„^i{ip)-»+^+ ai„{ip)-" 



A, 



hAip}"-^+h,(ip)"-^ + . . .+62„-i(>»- 



'-For a rcceiil slaluiiicnl of llio energy coiidilioiis in this form see I.. Boulliillon, 

 Revue Cenerale de I'Electricilc, 11, 1922, pages 656-661. 



"A nccessarj' and sufficient rondition that the real qiiaihatic form in 11 \arial)les 



2; i; «,^»>v,. 



j-\ k-\ 



(fly,. =0^,), 



be positive for all real x.dnes of the variables is that each of the >i ikleriiiinants 



On, 



(111 (ii> 

 "ll Oil 



flu ai2 . 

 flsi (Is: . 



a. I a.) ■ 



, a„„ 



be positive. For a proof of this see, for example, II. Hancock, "Theory of M.ixima 

 ,.nd Minima," 1<M7, iwges 82-91. 



