134 Transactions of the Royal Canadian Institute 



have included under it a specific function of the form of the second factor 

 in the product 



where w*'~^ is the highest power of u which presents itself under the 

 double summation and where we have a coefficient a'p_i_ ^_i + 0. 

 The principal residue in the product (5) is /3_p, „_<,a'p_i_ ^_i, 

 which is not identically 0. If, then, the principal coefficient in the 

 product (4) is identically for arbitrary values of the 2ni coefficients 

 ^-r^n-i the 2ni coefficients a;,_i, <_i must all vanish. It is immediately 

 apparent that on equating to turn about the principal residue relative 

 to 2 = in the product of the second factor in (4) by each of any 2ni 

 specific linearly independent functions contained under the form of the 

 first factor in (4) we impose on the 2ni coefficients a^_i, ,_i as many 

 linearly independent conditions. If, therefore, we condition the first 

 factor in (4) by the order numbers n, . . . ., r^, thus imposing X conditions 

 on the coefficients /3_y, „_^ and thereafter equate to the principal 

 residue in the product, we impose precisely 2ni — \ conditions on the 

 coefficients a^-i, /_i . 



We shall now assume for the moment that the equation (1) is of 

 ntegral character relative to the value s = 0. It is then evident that on 

 equating to the principal residue relative to s = in the product 



•-v-i 



6. 2-'((z, m)), S S a,_i,,_i2 



<=l,r=-«+l 



we impose on the coefficients a;._i, ^_i the 2ni—\ conditions referred to 

 above, for terms in the first factor of (6) involving z to as high a power as 

 s* cannot make any contribution to the principal residue in the product. 

 Assuming i to be taken sufficiently great the 2ni — \ conditions here in 

 question are precisely those conditions which are requisite in order that 

 the second factor in (6) may have the order numbers 7], . . ., r^ for 

 the value s = 0. These, then, are also the conditions which are imposed 

 on the general function of the form z~'{{z, u)) by the order numbers 

 Ti, . . . , Tr- The numbers of the conditions imposed on the general 

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

 and Ti, . . .,7;. respectively are therefore X and 2ni — \. The differ- 

 ence between the numbers of these conditions is 2\ — 2ni, and 



r 



this difference is readily seen to be also S (t^— Ts)vs- For if /i, . . . , t, 



s=l 



be a set of coincidences on a lower level than either of the 

 sets Ti, . . ., Ty or Ti, . . ., Tr and if i be chosen sufficiently great we 



