352 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
We then impose on the coefficients of the function {z, u) just those conditions which 
are necessary and sufficient for adjointness relative to the value z = a on equating to 
0 the residue relative to this value of 2: in the principal coefficient of each one of the 
products 
0./'^ (g, 'li) 
{z-aj 
'h (2, u ); 
,s = 1, 
( 22 ) 
That the conditions on the coefficients of the function d' {z, u) which we have 
just obtained are linearly independent of one another may readily be seen. For if 
the residues relative to the value 2 = a in the principal coefficients of the Ip^ products 
(22) were connected by a linear relation with constant multipliers, the linear 
expressioii in the functions 0/'^ (2, u) with the like multipliers would be a function 
(2, u) such that the residue relative to the value 2 = « in the principal coefficient 
of the product 
0^'^ {z, u) 
{z — aY 
"F (2, u) 
( 23 ) 
woidd be 0, no matter what the coefficients of F (2, u) might happen to be. The 
function 0^*^ (2, u) cannot vanish identically, since by hypothesis the functions 
0/'' (2, u) are linearly independent of one another. Suppose to be the highest 
power of u which appears in the expression of the function 0^*^ (2, u), and siippose, 
furthermore, that a term [3 {z — aY~'' actually presents itself. On choosing for 
F(2, u) the fnnction a{z — ay ^ ^ ® the residue of the principal coefficient in the 
product ( 23 ) will evidently be / 3 a, and this residue is not equal to 0 unless we have 
a = 0. There does not exist a function (2, m) then such that the residue relative 
to tlie value z — a in the product ( 23 ) is equal to 0 independently of the values of the 
coefficients of F (2, n). It follows that the I equations in the coefficients of the 
function F (2, v 3 ) obtained on equating to 0 the residues relative to the value 2 = a in 
the principal coefficients of the Ip^ products (22) are independent of one another. 
These equations, however, give the necessary and sufficient conditions for the adjoint¬ 
ness of F (2, n) relative to the value 2 = a, and we therefore have 4 = A- From (20) 
we then derive 
^ = I" ^ (ms~13- .(24) 
s = 1 \ vj 
For the number of the independent conditions which are imposed on the coefficients 
of the general integral rational function F (2, n) by a set of orders of coincidence 
ju'i, which are adjoint for the value 2 = a, we obtain from (I8) and ( 24 ) the 
expression 
2 
S = 1 
P-'sV, — k. = 2 
s = 1 
s = 1 
■1 + 
( 25 ) 
Representing in the form ( 19 ) the general rational function <I>(2, n) conditioned by 
a set of orders of coincidence ti, ...jTr for fde value 2 = a and equating to 0 the 
