FOiniAL KKAL1ZA15ILITY TIllOOUY — I 221 



open-circuited terminals is a finite passive 2n-pole (n = 2) which has 

 neither an impedance matrix nor an admittance matrix. 



2.42 When one ehminates this kind of descriptive difficulty by fixing 

 his attention only upon 2/;-pol(\s for which an impedance matrix (or, 

 dually, an admittance matrix) is available, the general realizability 

 prol^lem for finite passive devices is solved. A partial solution, for the 

 case n = 2, was published by C. M. Gewertz in 1933. The solution 

 (Theorem 1.1) of the problem for a general value of n has been an- 

 nounced recently by three authors, independently: Y. Oono, in 194G,* 

 the present author, in 1948, t and M. Bayard,' in 1949. The problem 

 for reactive 2n-poles is much simpler and was solved by Cauer, in 

 1931. 



2.5 Intermediate between the fairly specific problems of filter theory on 

 the one hand and the general realizability theory of multi-terminal 

 devices on the other, lies the study of four-poles as transducers. There 

 is a considerable literature on the realization of transfer functions or 

 transfer ipipedances under various structural limitations. The basic 

 and extensive work of Bode^ on active systems belongs also in this 

 category since it is avowedly limited to single-loop structures. 



2.6 Beyond the important result that, by sufficiently elaborate circum- 

 ventions, one may avoid the use of transformers in the synthesis of any 

 two-pole, (Bott and Duffin ) little in general is known about networks 

 which do not have transformers. 



2.7 The present paper is an essay in the realizability theory of devices 

 with many accessible terminals. Ideal transformers are admitted as 

 network elements; indeed, their use is essential. This fact is indicated 

 by the adjective "formal" appearing in the title. 



The availability of ideal transformers makes it possible to exploit 

 the simplification noted by Cauer and to consider only networks which 

 are 2w-poles in his sense. The aim of the paper, therefore, is to set a 

 foundation for realizability theory which is completely general within 

 this framework. 



2.71 The first problem is that of description. We observed above an 

 example — entirely trivial — of a passive four-pole which had neither an 

 impedance nor an admittance matrix. Unfortunately, opportunities 



* Date of Japanese publication. This reference, and manuscript of Oono'"- ", 

 were sent by Professor Oono in a personal communication to R. L. Dietzold, who 

 showed them to me in December, 1948, while an early draft of the present paper 

 was in preparation. The recent (1950) American repul)licat ion <jf Oono'" unfor- 

 tunately omits reference to the original. 



t Cf. footnote to 1.1. 



