THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
367 
which are complementary adjoint to the order 2 to the orders of coincidence 
are obtained on equating to 0 the principal residue relative to the value 
2 = GO in the product 
(-, u) d' {z, w) = - 2 (-, w) . d' {z, u). 
\Z ) s=l \Z / 
Here {l/z,ii) is the general function of the form implied by the index, subject to 
the condition that it possess the orders of coincidence Furthermore, 
the index i exceeds by 2 at least the degree of ^ (z, v,) in ^ and is at the same time so 
large that a term involving z~", or a higher power of l/z, is unconditioned by either of 
the sets of orders of coincidence or 
Writing 
— P(z, u} + 
i-l 
q = l 
2 " 
y - —q, n—ir 
<lyn-t 
(71) 
we see that the necessary and sufficient conditions in order that d' ( 2 , may have for 
the value 2 = 00 orders of coincidence which are complementary adjoint to the order 2 
to the orders of coincidence ...,Ty*>are obtained on equating to 0 the principal 
residue relative to the value 2 = co In the product 
2 2 d^(2,w.).(72) 
t=l q=l 
Here the coefficients are linear In the arbitrary constants Choosing the 
integer i sufficiently large we shall now take this for the value also of each of the 
integersy,, above. For the constants c_g n_f in (66) we evidently have 
' — q. 71 —t 
= 2 ^ 
7 
M 
-q, n — t > 
q = 1,2, ; t = 1, 2, 
71 , 
■ (73) 
where the summation with regard to k is supposed to extend not only to the finite 
values 2 = r4, which appear in the double summation in (65), but where it is also 
assumed to contain the term The expressions are linear in terms of 
the arbitrary constants S corresponding to the finite values 2 = and the value 2 = 00 . 
On equating to 0 independently of the values of the arbitrary constants d the 
principal residue relative to the value 2 = qo in the product 
71 7 — 1 
2 2 c ,_iZ-^u’^-\'P{z,u) .(74) 
t = l q=l 
we evidently obtain the necessary and sufficient conditions in order that the integral 
rational function ’F ( 2 , u) of degree M in 2 should be built on the basis (t) com¬ 
plementary to the basis (t). The conditions so obtained namely coincide with the 
aggregate of the conditions obtained on equating to 0 the principal residue relative to 
the value 2 = co hi eacli of the products (70) and in the product (72). 
