THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
341 
the branch with the function {z,u). The order of coincidence of the branch 
= 0 with the factor tt —P« we shall briefly refer to also as the order of 
coincidence of the branch = 0 with the branch u-V^ = 0, and we shall indicate 
this order of coincidence by the symbol fx.^f If is evident that Further¬ 
more, Jx, is equal to the sum of the orders of coincidence of the branch w—P^ = 0 with 
its n—1 conjugate branches, and we therefore have 
Ms = Ms, 1+•••+Mj,s-1 + Ms,s + 1+•••+Ms,n 
(5) 
It is readily seen that the numbers corresponding to the several branches of the 
same cycle are all equal. The r numbers thus defined for the branches of the r cycles 
we shall indicate by the symbols 
Ml) M2) •••) Mr.(^) 
The functions Qs(2, in (4) are defined by the identities 
f{z,u) = (w-P,) s = 1, 2, ..., n .(7) 
We can then represent any function Y{{z,u), of rational character for the value 
z = a (or 2 = oo), in the form"^ 
H (2,'?x) = 0iQi{2,n) + ... + 0,Q,(2,n) + ...-t0„QJZ)M) .... (8) 
where 0^, ..., 0„ are series in powers of z — a (or I/ 2 ) involving integral or fractional 
exponents, of which a finite number only can be negative. The necessary and 
sufficient condition that the function H ( 2 , ic) should be represented by the expression 
on the right-hand side of (8) is 
n — IT ( 2 , P;) — .. „ 
Q,(2,P,)’"“'’^’ 
n. 
(9) 
To see this it is only necessary to note that the functions 
Qi (z, u), ..., Q,_1 ( 2 , u), ( 2 , m), ..., Q„ ( 2 , u) 
all vanish identically on substituting in them u = P^. The representation of the 
function 11(2, tt) in the form (8) then exists and is unique. This representation 
evidently also gives the function in its reduced form since is the highest power 
of u which presents itself. 
The order of coincidence of the branch w —P^, = 0 with the function H ( 2 , m) is 
plainly the same as its order of coincidence with the element 0sQs(2,'?<) in (8) and is. 
* This form of representation was suggested to the writer by formula (3) in Chapter XIII. of his book on 
the algebraic functions, already cited. It may be pointed out, however, that the same form was derived 
by Christoetel from Lagrange’s interpolation formula and employed in his paper, “ Algebraischei 
Beweis des Satzes von der Anzahl der linearunabhiingigen Integrale erster Gattung,” ‘Annali di Mate- 
matica/ ser. 11., t. X., pp. 81-100. 
