THEOEY OF ALGEBKAIC FUNCTIONS OF ONE VAEIABLE. 
355 
it may have the orders of coincidence indicated in (31) is given by the formula (32). 
This same formula, then, gives the number of the conditions which must be satisfied 
by the coefficients in the expression 
.(34) 
in order that it may have, for the value ^ = 0, the orders of coincidence indicated 
in (29). Also the general rational function of (^, >/), conditioned by the set of orders 
of coincidence (29), must be included under the form (34) since the general rational 
function of (^, >y), conditioned by the set of orders of coincidence (3l), is included under 
the form (33). 
Transforming the expression (34) to terms of ( 2 , u) we see that the general rational 
function of ( 2 , u), conditioned for the value 2 = 00 by the set of orders of coincidence 
(29), is included under the form 
ft,-, (i) + M-n ... +2”'—Vo(i).(35) 
Furthermore, it is obtained from this form on subjecting the coefficients to a succession 
of conditions whose number is given by the expression (32). 
Let us now consider, in its reduced form, the general rational function of ( 2 , v) with 
its coefficients represented as series iiL powers of I/ 2 . The general rational function 
so represented, whose coefficients Involve no exponent which is < — A. we shall indicate 
by the notation"^ (I/ 2 , m). Taking A ^ (n—l) the general function E,_;^(l/ 2 , n) 
will certainl)^ include all rational functions of the form (35) and will, therefore, in 
particular include all rational functions of ( 2 , u) which are adjoint for the value 
2 = 00 . To pass from the general function E,_;^(l/ 2 , u) to the general form given in 
(35) we must, for s = 1 , 2, ..., n, reduce the degree in 2 of the coefficient of in the 
function from X to In reducing the general function Il_x(l/2, w) to the 
form ( 35 ) then we impose on the coefficients of the function a succession of conditions 
whose number is given by the sum 
n 
2 {X—7n(s —1)} = {71 — 1) .( 36 ) 
S = 1 
To this number we evidently only liave to add the number given in (32) in order to 
obtain the total number of the conditions wliich we must impose on the coefficients of 
the general function (I/ 2 , u) in order that it may be adjoint for the value 2 = go. 
We therefore impose just 
7l\ + ^ 
2 
S = 1 
-1 + 
(cc) 
/ 
(00) 
(37) 
conditions on the constant coefficients in the general function Il_x (I/ 2 , u) in order that 
it may be adjoint for the value 2 = oc. 
* It may be noted that a suffix will have the significance here attached to - X only in connection with 
the letter R. 
2 z 2 
