Conditions Imposed by a Set of Order Numbers 137 



independent in the n{i—j) coefficients a^_i. t-\ in which r has one of the 

 values — j, . . . , — i+1. All the other conditions may be regarded 

 as imposed on the coefficients a^_i, t-x in which r > —j. It is plain then 

 also that the addition to the first factor in (4) of any term involving a 

 power of 2 higher than z*~^ can introduce no further condition on the 

 coefficients a^_i, t-i when we equate to the principal residue in the 

 product, the integer i being chosen sufficiently great, for the presence of 

 such term could impose no conditions on coefficients a^-i, t-i in which 

 r> —j. The 2ni — \ conditions imposed on the coefficients a;-_i, ^-lon 

 equating to the principal residue in the product (4) therefore constitute 

 the total conditions requisite in order that the second factor in (4) may 

 have the order numbers r^, . . . , t^. The formula for X then follows 

 as before. 



In what precedes there is nothing essentially new. There is simply 

 some slight modification and abbreviation of what the writer has already 

 given elsewhere. When however a fundamental formula like formula 

 (8) has been obtained in one way it is usually not difficult to arrive at 

 the result in a multiplicity of other ways. Knowing the formula in 

 advance we have our orientation and it is only necessary to consider 

 it in its various connections and to note its bearing on the elements 

 involved in order to anticipate a line of proof depending on these ele- 

 ments. We shall here adjoin a couple of proofs so obtained. 



§ 2, We know from elementary considerations that the number of 

 the conditions imposed on a rational function of {z, u) of sufficient 

 generality by a set of order numbers corresponding to the value 2 = 

 is given by the sum of these order numbers plus a fixed number which 

 is independent of the values of the order numbers. In particular the 

 number of the conditions imposed on the general function of the form 



r 



s~* ((2, w)) by the set of order numbers n, . . . , r^ is ni -\r '^ t^v^ — A 



5=1 



where ^ is a definite number the determination of which would give us 

 formula (8). To determine A it will then suffice to determine the number 

 of the conditions imposed on the general function of the form 2~* ((s, u)) 

 for a sufficiently great value of i by a specific set of order numbers. 



To abbreviate matters we shall suppose as before that the equation 

 (1) is integral. The general rational function which is adjoint relatively 

 to the value 2 = is included under the form ((2, u)). What is the number 

 of the conditions imposed on this general form by the set of adjoint 

 order numbers /xi — 1 + 1/1^1, . . . , ju^ — l + l/»'r? 



We shall emp'oy the notations 2~*((2, w))o and ((2, «))« to designate 

 respectively the general rational function of integral algebraic character 



