136 Transactions of the Royal Canadian Institute 



general form z~*{(z, u)) on equating ln(n — l)j coefficients in the latter 

 form to 0. The number of the conditions imposed on the general 

 function of the form z~'^{{z, u)) by the set of order numbers n, . . . , r, 

 is, therefore, W -{-^n(n — l)j. From (9) then we see that the number of 

 the conditions imposed on the general function of the form s~*((z, u)) 

 by the set of order numbers n, . . ., T;. is given by formula (8) whether 

 the equation (1) is integral or not. The formula (7) holds good at the 

 same time, too, as we see on combining (8) with (2). 



Formulae (7) and (8) have been obtained on the assumption that 



— i has been taken sufficiently small. There is a least number —i' such 

 that terms involving the power z~'may actually present themselves in 

 a rational function conditioned by the set of order numbers n, . . ., r^. 

 It is evident that our formulae hold good so long as we have taken 



— i < —i'. 



If for Ti, . . ., Tr we take the order numbers defining adjointness for 

 the value s = 0, formulae (7) and (8) evidently give us the expression 



s=l. 



for the number of the conditions imposed by adjointness relative to the 

 value s = on the general function of the form z~*{{z, u)), while the same 

 formulae give us the expression 



r 

 s = l 



for the number of the conditions imposed on the general function of this 

 form by the 0-set of order numbers corresponding to the value s = 0. 



The formulae obtained in the preceding can also be derived directly 

 in the case where the equation (1) is nonintegral. This we have already 

 shown elsewhere. We shall however here indicate it in abbreviated 

 form. We shall for the moment suppose the coefficients /3_^, „_^ and 

 ar-i,t-i in the product (4) to be arbitrary for r= —j, . . . , —i-\-l, the 

 remaining coefficients being 0. If we hold the n{i—j) coefficients ^-r,n-t 

 here in question arbitrary and equate to the principal residue in the 

 product the n(i—J) coefficients a^_i, t-i, in which r has one of the values 

 —j, . . . , — i+1, evidently all vanish. That is to say we subject these 

 n{i—j) coefficients to as many linearly independent conditions. 



In the case then where the 2ni coefficients a^_i, t-i are assumed to be 

 arbitrary to begin with, as also the n{i—j) coefficients ^-r,n-t for which 

 r has one of the values —j, . . . , — i+1, whatever restrictions may be 

 placed on the other coefficients /S-^, „_;, it is evident that among the con- 

 ditions imposed on the coefficients a^_i, t-\ when we equate to the prin- 

 cipal residue in the product (4), are n{i—j) conditions which are linearly 



