NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 41.S 
Consequently, the terms R- and R" of R„ corresponding to Xi in the same way as R' 
and R" in general to will become 
00 
Ri= + 
R"= — .jp^.Xi 
— w^-.+i} .%i 
- 1 + . . . . W^_2.+ , } . &c 
in which it will be recollected that z=x—i. 
(32.) Now if V be the greatest common measure of m and 6’ {v being 1, if these 
numbers be prime to each other), we have 
and consequently the value of R, or R^+R; becomes 
Ri — (~ • Xi • { w.. + • • • • Wr-s+ l}-Xi 
s— • • • •^x—2s+ ’Xi 
— &c. 
(33.) Assembling together similar results for Ro, Ri, .... R,„_i, we have 
^={Xo’f^^ + Xl’'^^-l+ Xm-l’V^-m+l} 
— +Xi’'>n^-1 +””Xm-i 
— {n^-i-....n^-2s+l}{Xo’'»^^-s +Xl’'>^h-s-l + ””Xm-l’mv-s+l} 
^^x—3s + l } {Xo’m^-2s+Xi’'^‘^-2s-i + ””Xm-i’'m 
j:— 2s+ 1 } 
— &c. 
Now because m and s have v for a common measure, and that n=ts is the first 
s s 
multiple of s, which is also a multiple of m, it follows that n=-.m, - being an integer. 
Hence we have by equation (22.), 
• • • • ~\~^x—n+m 
^x—l ^x~\~\~^ x—m—\~\~ ^x—n+m—l 
&c.=&c. 
Substituting these therefore, and so arranging the terms that shall always stand 
first, the series within the brackets on the right-hand, in the expression for Z, become 
respectively 
Xo’n^+Xi’nx-i-\- Xn-i-n^-n+i 
Xm-s’'nx-\-X m— s+ 1 • ^'X ‘- 1 + X m—s-\-n—\ n + 1 
Kjm—2s • %7tt— 2s + l • • • *')(pn — 2s+n — l ‘^.r— «+l ? 
