200 THE ROYAL SOCIETY OF CANADA 



8. Writing 



^(r(2,«)=/«w"-'^"^ + /«-iW«-<^-2+...+/^.^2^^+/^+j,o- = 0,l, . 

 formula (7) takes the form 



9. ' J/(a,w) =^'2 i 9s F(y{z,Ps )u'^ . 



cr = o 5 = 1 



In H{z, u) then the coefficient of u<^ is 



10. -E Os F^{z,Ps). 

 5=1 



Suppose now that the orders of coincidence of the function Fa (2, u) 



with the branches of the r cycles corresponding to the value z^a 



are designated by the symbols Mo-, l, Mo-, 2 »• • •» tJ-a,r respectively. If 



in the sum (10) the portion corresponding to the cycle of order vi 



is given by the sum 



11. i:'9sFa{z,Ps) 



s = l 



we see that the lowest exponent in the vi conjugate series 61, . . . , 6vi 

 consistent with this expression being integral relatively to the element 

 z — a is — iJL , — 1+1/fi. When however this is the lowest exponent 



(7, 1 



in the series di, . . ., dp^we see from (4) that the order of coincidence 

 of the function Hiz,u) with the branches of the cycle of order vi is 

 Hi — fjL , — 1 + 1/vi where m is the order of coincidence of the functions 



Qi(z,u), . . . , Qv^{z,u) with the respective branches w— Pi = 0, .. . , 

 w — P^j = 0. We see then that the orders of coincidence of a rational 

 function H{z, u) with the branches of the r cycles corresponding to 

 the value z = a which insure that the coefficient h^ of u^ in the 

 function is integral with regard to the element z — a are respectively 

 the numbers 



12. tJ-i-fJ-a,!— 1 + 1 Ai, ... , Mr — Mo-, '- — 1 + lA»- • 



These are evidently the orders of coincidence furnished for the value 

 2 = a by the function 



13 A{z,u) 



Fa {z, u) 



where the numerator represents the general adjoint function. The 

 form (13) is independent of the particular value z = a under con- 

 sideration and the argument holds also for the value 2 = °° , the element 

 z — a being here replaced by I/2. In the particular case where o- = w— 1 

 wehaveF„_i(2, «)=/„ = 1 and consequently mo-,1 = Mo-, 2= • • • =^0-, r= 0. 

 The numbers in (12) here become the numbers defining adjointness 

 for the value of the variable 2 in question and the quotient (13) 

 becomes the general adjoint function, all of which accords with what 

 we already know in this particular case. 



In the set of numbers (12) we evidently have just that set of orders 

 of coincidence relative to the finite value z = a which insures that the 



