370 On the Equation in Numbers of the First Degree. 

 we have N' = 2a; + N, or 



v,._v(a/ Aa. A( -n) ^ 



^'""-^^lO -A«f (1-A6) ... (l-A/) J • 

 In like manner, if we vvi'itc 



ax' + ax"->tax'"-\-by + cz+ ... -^-Iw^n, 

 the sohitions of this equation spring from those of equation (1) 

 by making a;' + .T"-|-a,"' = A', the number of sokitions of which 

 equahty is (.T + l)('2^ + 2) •• wherefore 



^^^+3^+3 _^r A(-n) \ . 



2 ""^ 1(1-A«)^(1--A6)...(1-A/)/' 

 from which w^e may readily deduce; by aid of what h as bee 

 already shown, 



v,.2_v/a (Aa)(l+Aa)A(-?i) . 



i.z -^^(l_A«)3(l_A^,)...(l_A/)' 



and so in general, 



(1-A«y+'(1-A6)...(l-A/)" 



Again, if we write 



ax-irhyi + byc^+ ...-\-by^-\-cz+ ... ■\-lw=n, . . (4) 



we shall find by parity of reasoning (seeing that in this last equa- 

 tion the solutions may be derived from those of equation (1) by 

 keeping x, z, ... iv all unaltered, whilst we give to ?/„ y^ ... y^ all 

 the values compatible with ?/i + ?/2+ ... +y^ = y), the value of 

 1,x' in equation (4) will be the same as that of 



(y + l)(y + 2)...(y + e) 

 ^"^•~ 1 . 3 ... e" 



in equation (1). Wherefore we shall evidently obtain 



__^^Aa{l + Aa)...({i-l+Aa}xAb{l+Ab)...({6-l)+Ab), 

 {l-Aay+\l-Aby^\l-Ac)...{l-Al) 



l,x\y' = 'Z@ ^^ ^ ^.^j 



the extension of the theorem to 2*' .y^.z"... is too obvious to 

 need further allusion. 



Thus, then, to find x^, x\ . . . x^, we may begin by forming 

 an equation of the Nth degree, whose coefficients are known, 

 because the sums of the powers of the roots are given. Sup- 

 posing these roots to consist of N, values x, N^ values Xc^, ... N^^i 

 values a-^, the solution of yu. simple equations will enable us to 

 find the sum of the N, values of y corresponding to x-^, the sum 

 of the Ng values of y corresponding to x\. , . , and the sum of 



