THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
371 
the letter s. In the case where s = 0 the constants ^ must all have the value 0. In 
this case, then, on referring to the representation given in (59), we see that the 
general rational function H ( 0 , u) built on the basis (r) must reduce to the polynomial 
V{z,u), and on referring also to the identity (65), we furthermore see that the 
polynomial P ( 0 , u) must be 0 identically since its coefficients are linear in terms of 
the constants Where S = 0 then the only rational function built on the basis 
(t) is the constant 0. 
Where s is >0 we can select among the constants (i a set of s arbitrary constants 
(ii, .,.,(is, in terms of which we can express all the other constants 8 linearly. The 
representation of the general rational function of ( 0 , u) built on the basis (t) will be 
obtained in the form (59) on replacing in the functions {z,u) each of the constants 
by its linear expression in terms of the arbitrary constants ^ 1 , ..., and on doing 
the same for each of the constants which presents itself in the coefficients of 
P ( 0 , u). The general rational function built on the basis (t) can then be represented 
in the form 
^iUi + ...+^sUs.(79) 
where Ui, ..., Us are specific rational functions of ( 0 , u). 
That the functions Ui, ..., Ug here in question are linearly independent of one another 
we can readily show. For suppose, if possible, that these functions are connected by 
a linear relation 
c?iUi +... + (igUs = 6.(80) 
We have seen that the constants S must all be equal to 0 if the function represented 
by the forms (48) and (59) is to be 0 identically. Now the form (79) is obtained from 
the form (59) on expressing each of the remaining constants 8 in terms of the s 
constants ^ 1 , ..., 8s, and the left-hand side of (80) is thereafter obtained from the form 
( 79 ) by giving to the constants (ii, ..., 4 the values respectively. The 
left-hand side of (80) is then obtained from the form (59) on attributing to the 
constants 8 in this form certain values, including the values di,...,ds for the 
constants (ij, ...,4, respectively. The resulting function, however, is identically 0, 
and consequently d^, ...,ds must all be 0. The functions Ui, ...,Us, then, are not 
connected by a linear relation involving multipliers which are different from 0. The 
general rational function built on the basis (r) then involves effectively s arbitrary 
constants as we see from its representation in the form (79). Employing the 
notation N,. to designate the number of the arbitrary constants involved in the 
expression of the general rational function built on the basis (r), we have from (78) 
N, = N;-fn-2 S T, 
K S = 1 
Wi iv 
G +2^ 
2 
K s = 1 
M, 
(0 
-1 + 
M 
. . (81) 
In deriving this formula the sole limitation on the basis (r) was that all of its 
numbers corresponding to finite values of the variable 0 were zero or negative. 
3 B 2 
