144 Transactions of the Royal Canadian Institute 



((s, w)) takes place. As before we shall understand that this function 

 is of degree n — 1 in u and /(z, u) we shall assume to be a product of r 

 cyclical factors /i (2, w) , . . . ,/r(s, w)- 



Let us consider a set of order numbers n, . . . , t^. which are adjoint 

 but not simultaneously greater than the numbers /xi, . . . , ju^. We shall 

 suppose Ti , . . . , Ty_i to have values as great as we will and t^ will then 

 have one of the values ^ff— l + l/j'rt • • • . Mr- These order numbers we 

 propose to impose successively in the order ti, t2, ..... t^ on the function 

 {(z,u)). They will not require the principal coefficient to be divisible by 

 z and they will evidently impose the same number of conditions on the 

 general function {{z,u)) which is unrestricted to begin with and on the 

 general function after we have assigned a specific form to the principal 

 coefficient so long as the form in question is not divisible by z. We shall 

 then from this on assume that the otherwise general function {(z,u)) 

 has 1 for its principal coefficient. 



In accord with formula (17) the order number ti imposes 

 Ti j/i — I (/x/ — 1 + 1/1/1) vi conditions on the coefficients of the general poly- 

 nomial of degree vi— 1 in w with coefficients of integral rational character 

 for the value z = 0. This then is also the number of t he conditions imposed 

 by the order number n on the general polynomial of degree n — l^vi—1 

 in ti, with coefficients of integral rational character relative to the value 

 2 = 0, and therewith the number of the conditions imposed by this order 

 number on the function ((z, w)) here in question. We may now suppose 

 this function to be represented as a product ((s,w))^^ ((z, «))«-•' -1 where 

 the suffixes indicate the degrees of the respective factors in w, the co- 

 efficient of the highest power of u being 1 in the case of each factor. 

 The two factors here in question are not uniquely characterized. If we 

 suppose the first factor to contribute the total order number n and the 

 second factor to be arbitrary, save for the coefficient of u*'~''^~^ the pro- 

 duct is included under the function ((z,w)) conditioned by the order 

 number ri. Included under the function so conditioned however are 

 also products in which the first factor does not contribute the total 

 order number ti. In any case we may assume ri to be so great that the 

 first factor must coincide with /i(z, u) out to such point, and as far 

 beyond as we will, that its orders of coincidence with the branches of the 

 equations /2(z, z/) = 0i • • • ./r(2, w)=0 are the same as those of the func- 

 tion /i(z, u) with these branches. 



We shall for the moment employ the notation )Uj,/to designate the 

 order of coincidence of a branch of the tth cycle with the 5th cyclical 

 factor /j(z, w) of/(z,M). Let us now split off from ((z,w))„_^^_i a factor 

 ((2,«)),j of degree V2 in u on imposing on the former function the order 



