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XLI. Note on the Equation in Numbers of the First Degree 

 between any number of Variables ivith Positive Coefficients. Bij 

 J. J. Sylvester, M.A., F.R.S., Professor of Mathematics in 

 the Royal Military Academy*-. 



I PROPOSE to show that all the systems of values {a;,y,z...w) 

 which satisfy a given equation in integers, 



ax + by + cz+ ... +ho = n, 

 ({a,b,c...l) being all positive, and the number of systems 

 therefore definite), may be made to depend on algebraical equa- 

 tions whose coefficients are known functions of «, b, c.l and n. 

 The fact is somewhat surprising, the proof easy, being an imme- 

 diate consequence of the theorem I have given in the Quarterly 

 Journal of IMathematics, and also in Tortoliui's Amiali for Jan. 

 1857, of the problem of the partition of numbers. 



For my present purpose, this theorem may be with advantage 

 presented under a somewhat modified form as follows: — Let 



%{¥t) be used to denote the coefficient of - in the expansion of 



Yt in ascending powers of t. Let N stand for the number of 

 solutions of the equation 



ax + by + cz+ ... +lw = n; 

 let m be the least common multiple of a, h, c,...l, 

 p be any primitive root of p'" = l, 

 and pe-P* be called A;^ ; then 



^-^^l{i-Aa){l-Ab)...{l-Al)j 



If now we call N' what N becomes when, in lieu of the 



equation 



ax + by + cs+ ...+ho = n, (1) 



ax' + ax" + by + cz+ ...+lw = n, . . . (2) 

 it is clear that 



But it is also clear that all the solutions of equation (2) may be 

 derived from those of equation (1), by writing for each value of x 



x' + x" = x; (3) 



and as the number of solutions of equation (3) is evidently x + 1, 



* Communicated by the Autlior. 

 Phil. Mag, S. 4. Vol. 16. No. 108. Nov. 1858. 2 B 



we write 



