118 THE ROYAL SOCIETY OF CANADA 



furnishes a process, involving rational operations only, to determine 

 whether the equation (1) is, or is not irreducible in the case where 

 this equation is integral of degree n in (z,u), the terms of degree n 

 splitting up into n distinct linear factors. The process also deter- 

 mines the number p of the irreducible equations involved in equation 

 (1.). This process would involve first the construction of a basis 



(o) (1) (n-\) 



A (z,u), A (z,u), ,A {z,u) for the rational functions of 



integral algebraic character. The coefficients B , B\, ,B n -\ 



in the general polynomial B{z,n) of degree n— 1 in (z,u) we should 

 then subject to the conditions implied in giving integral character 



<*> 

 to the w— 1 expressions C n -\ represented in (9.). In the resulting 



polynomial B(z, u) we should equate to o the coefficients of all terms 

 of degrees n — 1 and n — 2 and note the number of the further conditions 

 so imposed on the constant coefficients of the polynomial. This 

 number would be equal to In — p. 



If our equation (1.) is not of the character which we have had in 

 view in what immediately precedes we can readily transform it to a 

 form which will suit our purpose. If it is not already integral, a 

 simple transformation of the dependent variable, we know, will give 

 us an integral equation. We shall then assume that the equation 

 (1) is integral. If the character of the equation for the value z— «= 

 does not suit our purpose we may readily transform to one which will. 

 That is to say we select any value z = a to which correspond n different 



values of u. These values we shall indicate by (Si , , P„ . We 



do not need however to actually solve the equation whose roots we 

 here indicate. Putting 'Ç = (z — a)'\ v = Ç a u, where a is a properly 

 chosen integer, the equation (1.) transforms to an integral equation 

 F(Ç, v) =o. The n series giving the values of v in the neighborhood 

 of f = co evidently begin with the terms /3i i' a ,. . . . , /3 n Ç a respectively. 

 The n orders of coincidence requisite to adjointness relative to the 

 equation F{Ç , v)=o for the value f = » evidently all have the same 

 value — {n — I) a. On regarding f as of dimension 1 and v as of dimen- 

 sion a and on representing the general rational function of (f, v), 

 which is adjoint for the value f = °= , in the form 



10. B(z, v)=B n - l v"- 1 +B n -2V»-~+ +B , 



it is readily seen that this function is of dimension (n — l)a. Other- 

 wise said the degree in f of B„- r is (r— l)a for r= 1, 2, , n. 



To determine then for the equation F(i',v)=o, and there- 

 with for the equation f(z,u) =o, whether it is reducible or irreducible, 

 we begin by constructing a basis for the rational functions of (f, v) 

 of integral algebraic character relative to the equation F(Ç,v)=ô. 

 With the aid of this basis and by the means indicated in the earlier 

 part of this paper we impose on the general integral rational function 

 B(Ç,v) of dimension (n — l)a the conditions requisite to adjointness 

 for all finite values of the variable f . We thus obtain the general 

 adjoint function associated with the equation F(Ç,v)=o. In this 



