354 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
the numbers will then be 0 or positive. To say in this case that a rational function 
of (2, u) is conditioned by the set of orders of coincidence tj, for the value z = a 
is equivalent to saying that it becomes infinite for the branches of the several cycles 
corresponding to the value z = a to orders which do not exceed the numbers cti, o-, 
respectively. 
§ 4 . We shall now consider the connection between the form of a rational function 
of {z,u) and its orders of coincidence for the value z = zc. Indicating by the 
number of the cycles of the equation (l) for the value 2 = 00 and by the 
orders of these cycles, we represent by the notation 
Ml 
(oc) 
-1 + 
(00) ’ 
Mr, 
(oc) 
-1 + 
(=d) 
( 29 ) 
the orders of coincidence which define adjointness for the branches of the several 
cycles. On introducing two new variables, f = 2~b >] — z~"'u, where m is a properly 
chosen integer, the equation (l) goes over into an equation 
^ + >--+5'o — O 5 .(30) 
in which the coefficients are integral rational functions of Bational functions of 
(f, 7 ]) are rational functions of (2, ?i), and conversely. The branches of the equation ( 30 ) 
for the value $ — 0 correspond individually to the branches of the equation (l) for the 
value 2 = GO and group themselves in like manner into cycles of orders 
respectively. Also it is evident that adjointness relative to the equation ( 30 ) for the 
value ^ = 0 is defined by the orders of coincidence 
m (m— l) + ;u/°"^ —1 H—Ml (li—+ —1 + —. . . (31) 
»'i '^>•00 
obtained on adding l) to each of the numbers given in ( 29 ). The general 
rational function of (^, >/), which is adjoint relatively to the equation ( 30 ) for the value 
^ = 0 , is integral with regard to the element ^ since the equation is an integral 
algebraic equation. Furthermore, on referring to formulae ( 24 ) and ( 3 l) we obtain 
immediately the expression 
+ J 2 .(32) 
S=1 \ D /- 
for the number of the conditions which must be imposed on the coefficients of the 
general rational function of (f, >/),, of integral character for the value ^ = 0 , in order 
that it may be adjoint relatively to the equation ( 30 ) for this value of the variable. 
Ptepresent the general rational function of (^,»/), of integral character for the value 
f = 0 , by the expression 
pn-\ (f ) V" +p„_2 ii) . .. +/)o (f).(33) 
The number of the conditions which must be imposed on its coefficients in order.that 
