[fields] adjoint orders OF COINCIDENCE 251 



with the n branches u — Pi = Q, . . . , u — P„ =0 are therefore greater 

 than the numbers /Ii — 1, . . . , iu» — 1 respectively. 



We shall now rearrange the series Pi, . . . , Pt , indicating the new 

 arrangement by the notation P^i» -f «2» • • • » ^nt • The notation is 

 here so chosen that each of the t branches u — Pn^ =0 has with the 

 corresponding factor u — Qi in the product (8), an order of coincidence 

 which is at least as high as the order of coincidence of any of the t — i 

 branches u — Pti-_^^ = ^, • • ■ , u—^Kt =0 with this factor. We further- 

 more replace successively in the product (8) the factors u — Qi, . . . , 

 u — Qi-i by the factors u—P^^, • • • , u — P^f_^. At no step in the 

 process do we diminish the order of coincidence of the product with 

 any of the n branches. The orders of coincidence of the product 



(9) {u-PKy) • ■ . {u-Pk^-Ù {U-Qt) {U-Pt + X) . . . {U-P„) 



with the n branches u—Pi = 0, . . . , u — P„ =0 are therefore greater 

 than the numbers jûi — 1, . . . , /T» — 1 respectively. 



In particular, the order of coincidence of the product (9) with 

 the branch u—Pk^=0 is greater than JI^^—1, and therefore the 

 order of coincidence of the factor u — Qt with this branch must be 

 greater than —1. The order of coincidence of the factor u — Qt with 

 the branch ii—Pt =0 was however, as we saw.at least equal to its order 

 of coincidence with any of the branches ii — Px = 0, . . . , « — P/-]=0 

 and therefore not less than its order of coincidence with the 

 branch u — Pn^^=Q. The order of coincidence of the factor ii — Qt 

 with the branch u—Pt =0 must then be greater than —1. We can 



consequently write Qt =Pt -{-{z — a)'^' St , or Qt =Pt -\-{l/z)^^ St where 

 a<is> — l,and where 5/ is a seriesinpowersof 2 — aor 1/ s none of whose 

 exponents is negative. The argument holds where t has any one of 

 the n values I, . . . , n. The identity (4) therefore takes the form 



(10) f{z,u)-\-H{z,u)={u-Pi-(z-a)'''S,} ... {u-P„-{z-a)'^nS„} 



or «. 

 (10a) fiz,u)-\-H{z,u)^{it-Pi-{l/z)'''Si} . . . {u-P„-{l/z)°-nS„} 



in the case where H{z, u) has orders of coincidence with the n branches 

 w — Pi = 0, . . . , « — P„=0, which are greater than the corresponding 

 numbers /Il — 1 , . . . ,/î„ — 1 respectively — the exponents a/ being > — 1, 

 and the series St involving no terms with negative exponents. 



On effecting the multiplications indicated on the right-hand side 

 of the identities (10) and (10a), the products can evidently be written 

 in the form.s 



(11) {u-Pi) . . . («-Pn) + (z-a)«Gfs-a, u) 

 and *■ 



(11a) (//-Pi) . . . {u-P„) + {\/zfG{\/z, u) 



where the exponent a is > — 1, and where in G{z — a,u) and C{l/z,u) 

 the coefificients of u"~'^ are integral relatively to the element z — a and 



