NUMBER OF IMPEDANCES OF AN n TERMINAL NETWORK 309 



n 



1=0 



All algebraic operations on umbral symbols are carried out as in ordi- 

 nary algebra except that the degrading of subscripts must not be 

 performed until operations are completed. It must be noted that 

 5" = 5o, hence is unity only when 5o = 1, as in the present case and 

 not always as in ordinary algebra. 



The umbrae for the impedance numbers are written D, T and U, 

 and by use of the generating identity above have the following gener- 

 ating identities: 



exp tD = i(e' - 1)2 exp (e' - 1), 



exp tT = i[4(e' - 1)^ + (e' - 1)*] exp (e' - 1), (6) 



exp tU = |[20(e' - 1)^ + 10(e' - 1)^ + (e« - 1)«] exp (e' - 1). 



These follow immediately from the base generating identity and 

 the factorial expressions for dx, t^ and Ux- 



Expanding these expressions in powers of e' gives alternate ex- 

 pressions as follows: 



exp tD = Ke^' - 2e' + 1) exp (e' - 1), 



exp tT = i(6*' - 6e2' -j- Se' - 3) exp (g« - 1), (6.1) 



exp tU = i(e«' + 4^5' - ISe^" + SSe^' - 36e' + 11) exp (e' - 1). 



To recapitulate, these expressions mean that Z>„, Tn and Un are 

 the coefficients of /"/w! in the expansions of the right-hand sides; 

 taking Dn, for example, the first equation of (6) is equivalent to the 

 equation : 



Dn^Um—Bie^ - 1)2 exp (e' - 1)], 

 <=o ^^ 



which may be shown to be equivalent to the first of equations (2). 



The generating identities lead immediately to recurrence relations, 

 as will now appear. 



Recurrence Relations 



Recurrence relations to be derived are all obtained by differentiation 

 with respect to /. Under this operation umbrae behave like ordinary 

 variables; thus 



-r exp tD = D exp tD 



= Di + D2t + D^^^+ ••• ^n+i^^+ •••, 



as may be verified readily. 



