250 THE ROYAL SOCIETY OF CANADA 



(5) fû{z,tO = ^i'<-PÙ ■ • • («-P/-l)("-P. + l) . . . (U-Pn) 



t I _ 



with the n branches we shall indicate by the symbols m, . . . , fi» 

 respectively. P2vidently )u^ is also the order of coincidence of the 

 product 



(6) (i/-P>) . . . (u-Pi-i) (u-Prn) . . . (n-P„) 



with the branch u — Pi = 0. The n branches group themselves into a 

 number r of cycles, whose orders we shall indicate by the symbols 

 v\, . . . , v^ respectively. The orders of coincidence of a rational 

 function with the branches of a cycle will all be the same. The orders 

 of coincidence of the function fû (2, it) with the branches of the r 

 cycles we shall indicate by the symbols ^i, . . . , fXr respectively. These 

 numbers, with repetitions it may be, will then in some order coincide 

 with the n numbers /Ti, . . . , ]!« • Orders of coincidence with the 

 branches of the several cycles which do not fall short of the numbers 



(7) /;i-l+— ,M. -1+ — 



Vi V,. 



respectively, we designate as adjoint orders of coincidence. If the 

 orders of coincidence of a function with the branches of the several 

 cycles corresponding to a given value of the variable z are adjoint, 

 we say that the function is adjoint for the value of the variable in 

 question. To say that a rational function of (s, u) is adjoint for a 

 value of the variable 2, is evidently equivalent to saying that its orders 

 of coincidence with the branches of the several cycles are greater 

 than the numbers /ii — 1, . . . , jUr — 1 respectively. This also amounts 

 to saying that the orders of coincidence of the rational function with 

 the n branches ii — Pi = Q, . . . ,ti — Pn —^ are greater than the numbers 

 Jul — 1, . . . , ]x„ — 1 respectively. 



We shall now consider a rational 'unction which is adjoint for 

 a value z = a (or 2= <»). Representirig the function in the form (4), 

 we may suppose that we have assigned the sulfixes of the P's, so that 

 the branch u — P„ =0 has at least as high an order of coincidence with 

 the factor u — Q„ as has any of the other n — 1 branches, so that 

 w — P„-i = has at leastashighanorder of coincidence with the factor 

 u — Qn-i as has any of the n — 2 branches u — Pi = 0, . . . , z/ — P„ - 2 = 0, 

 and in general so that the branch u—Pt =0 has at least as high an 

 order of coincidence with the factor u — Qt as has any of the t — \ 

 branches « — Pi = 0, . . . , « — P/-i =0. Choosing our notation then in 

 the manner just indicated, each factor n — Qt in the product on the 

 right-hand side of (4) is coordinated with a corresponding branch 

 u — Pt =0. In the product in question we now successively replace 

 the factors u — Q„, ii — Q„-i, . . . , « — G+i t>y the factors u — Pn, 

 u — Pu-\, . . . , w — P/+1. At no step in the process do we diminish 

 the c rdcr of coincidence of the product with any one of the n branches. 

 The orders of coincidence of the product 



•-«v 



(8) (m-O,) . . . (^u-Qt ) (u-Pf+i) . . . {ii-Pn) 



