372 
DE. J. C. FIELDS ON THE FOUNDATIONS OF THE 
Dropping this restriction, we shall now suppose (r) to be any basis whatever. The 
complementary basis, as before, we indicate by (f). The most general rational 
functions built on these bases we shall designate by H {z, u) and H {z, u) respectively. 
On properly choosing a definite polynomial g (z) it is readily seen that H (z, u)lg{z), 
and g (z) H (z, u) are the most general rational functions built on bases (^) and (^) 
which are complementary, the former basis at the same time offering no positive 
orders of coincidence for finite values of the variable z. We therefore have for the 
bases (^) and (^) the formula 
N, = N7+n—2 2 t 
(0 ( k ) , ly 
K S = 1 
2 . u. 
= 1 \ 
+ 
, (0 
(0 
Here, however, we evidently have = N,., Nj = N;, and 
n 
2 / 
«= 1 
M„(-<) _ 
= 22 
M 
; = 1 
so that we immediately verify the formula (81) for the more general complementary 
bases (t) and (r) here in question. 
From (57) and (58) we derive 
2 i t / V ’+2 2 f,<V’ = 2 k +2 2 + 
KS = 1 K S=l K S = l \ 
(k) 
• ■ ( 82 ) 
Combining this with (81) we obtain 
N, + i2 2 
t;V = N;+12 2t/V\ 
S = 1 K s = 1 
(83) 
This is the Conigilementary Theorem. The complementary theorem then states that 
the number of arbitrary constants involved in the expression of the general function 
built on a basis (r) plus half the sum of all the orders of coincidence required by the 
basis is equal to the like number constructed with reference to the complementary 
basis (f), that is, with reference to the basis whose numbers are connected with those 
of the basis (r) by the relations (57) and (58). 
The formula (83) continues to hold good when we replace (57) and (58) by the 
somewhat more general relations 
T“ + T« = m.“> -1 + = ; s = 1.n, 
and 
(oo), -(OO) (a) , 
. . . (84) 
. . . (85) 
where m/"', ..., mr^"^ represent the actual orders of coincidence of an arbitrarily chosen 
rational function E, (z, u) with the branches of the several cycles corresponding to the 
