[fields] an algebraic EQUATION 201 



{n — a)tli residue is in any rational function conditioned by these 

 orders of coincidence. If the numbers (12) are supposed to have 

 reference to the value 2= 0° we obtain, on adding 2 to each of them, a 

 set of numbers 



14. Ml - /io-, 1 + 1 + 1 Ai . • • • , Mr - Mo-, r + 1 + l/^r 



such that any rational function possessing these as orders of coinci- 

 dence relative to the value 2=°° evidently has as its {n — <j)th 

 residue relative to this value of z. We conclude then that the sets of 

 orders of coincidence corresponding to any specified value of 2, the 

 value 2= °° included, which insure the vanishing of the 1st, 2nd, 

 . . .«th residues respectively for the value of the variable 2 in question, 

 are furnished by the functions 



ij, (a, u) 4>{z,u) <j){z, u) 



Fn-liz,u) • Fn-2iz,u) ''•' Fo {z, u) 



where 4>{z,u) is the general ^-function. From the form of Fg- (2, u) in 

 (8) we see that we have 



16. wo-+iF^ (2, «)+/a«^ + . . . +/o = 



so that instead of the functions Fa (2, u) appearing in the denominators 

 of the expressions in (13) and (15) we might employ the forms 



17. -u-<^-^{Uu(^ + ...+fo). 



The orders of coincidence defined by the general functions 

 A{z,u)/F(^{z,u) we shall call quasi adjoint orders of coincidence and 

 the bases of coincidences on which these functions are built we shall 

 call quasi adjoint bases. For the value a = n — l then the quasi 

 adjoint basis coincides with the adjoint basis. In the reduced form 

 of the quasi adjoint function 



18. A (2, u)/F^ (2, u) = /^C^) w" -1 + . . . +a(^) 



n-l o 



we see that h^ , the coefficient of u , must be a constant for the orders 



of coincidence defining the function require that the coefficient of u^ 

 be integral with regard to every element z — a and also with regard to 

 the element I/2. The function A (2, u), and therefore also the function 

 A{z,u)/ F(j{z,u), involves* p-{-2n — p arbitrary constants where p 

 is the number of the irreducible equations involved in the funda- 

 mental equation (1). If in the general function (18) we add 1 to each 

 of the orders of coincidence corresponding to some one specific value 

 of a we impose n conditions on its constant coefficients. Among these 

 n conditions is evidently included the vanishing of the constant 

 coefficient of u<^. _ 



Suppose Ti, . . ., Tr and ti, . . .,T^ to be two sets of orders of 

 coincidence relative to the finite value z = a which are connected by 

 the relations 



* Theory of the Algebraic Functions of a Complex Variable, p. 150, Mayer & 

 Muller, 1906. 



