358 
DR. J. C. FIELDS ON THE FOUNDATIONS OF THE 
from the former function to the latter function we should have, for s = 1, — to 
reduce the degree of the coefficient of u" from A to \ — s. This would impose on the 
coefficients of the function Ji_^{l/z,u) in all ^n{n — l) conditions. Now we have seen 
in § 1 that the degree of a rational function of {z, u), which is adjoint for the value 
2 = CO, must be ^ N — 1 . Taking X = N — 1 , formula ( 37 ) gives us 
«(N-l) + i 2 L(">-i+U)i',‘"’ 
S = 1 \ ''s / 
for the number of the conditions which must be imposed on the coefficients of the 
general function lX_i^^i{lfz, u) in order that it may he adjoint for the value 2 = oo. 
Among these conditions are included the —l) conditions requisite to reduce the 
general function here in question to degree N — 1 . For the number of the conditions 
whicli must he imposed on the coefficients of the general reduced rational function of 
{z, n) of degree N— 1 , in order that it may l)e adjoint for the value 2 = oo, we then 
obtain the expression 
w (N-l)-|n(n-l) + ^ 2 
■s = 1 
(oo) 
-1 + 
(*) / D 
(ao) 
( 42 ) 
More generally, on subtracting l) from tlie expression given in ( 41 ), we 
obtain 
.9 = 1 
,, (®)_1 _L 
2 •^ \ ^ (cc) / *'s 
s = l \ Ps 
(^) 
■ . ( 43 ) 
for the number of the conditions imposed on the coefficients of the general rational 
function of (2, 71) of degree A by the set of orders of coincidence where A 
is not less than the greatest degree which a rational function can have and yet 
possess these orders of coincidence. 
The general rational function of (2, 71), whose coefficients are of degree A in 2, we 
shall represent in the form 
= .( 44 ) 
where in the first element the index {i) signifies that in the coefficients, arranged 
according to powers of I/2, the highest power which may appear is (1/2)*“^ while in 
the second element the notation ((1/2, n)) signifies a reduced polynomial inwhose 
coefficients, expanded in powers of I/2, present no negative exponents. Takiiig 
i sufficiently large and imposing on the function ] 4 _a( 1 / 2 :, 7 i) the orders of coincidence 
t/**, ..., for the value 2 = 00 the coefficients of the second element in the sum on 
the right of ( 44 ) will be unaffected. The number of the conditions to which the 
coefficients of the function li_;^(l/2, n) are thereby subjected, and therefore the 
number of the conditions imposed on the coefficients of the function u) by 
the orders of coincidence here in question, is given by the expression ( 41 ) on assuming 
