NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 415 
and therefore 
For Y we have 
'F(s) = 0.3,+ 1.3._,+|.3.., 
r(s)=0.3,+i.3..,+3,., 
whence by equation (37.) 
For Z we have 
whence 
Y=-i{2.6,_.+6.6,_,+2.6„,+6.6,., 
Q.= -i.2._, Q,.,= -h 
m=2, s=3, ms=6, v=l, Xo=- 2 ’ Xi = ^ 
and by (45.), (46), (47-), 
Z — — ^ • b.r-2 +3 . 6 j,_3+ 2 . 6^_4 + 1 . 6,,_5|- 
+ T^|64,+ 6^_2| 
= — 2.64,_4+5 .6^_5| 
adding all which parts together, we have X+Y+Z, or 
n(x) =^| - 6^_1 — 4 . 6^_2 + 3 . 6 ^_ 3 — 4 . 6,_4 — 6 ^_, 
(36.) When s=4, we have, therefore. 
and, therefore, by equation (35.), 
48^ 
For Y we have 
^(^)=I2’ ‘P'(^) = T2’ ‘P"(^) = 12 
x[x + 4){x-\-S) ^ x{x + 4:) 
x=^ix(x+3y — ^^.2(^+3)+ r" ' ~r" ' '^' . 2 - 
•1.2 
(s-ir 
12 
9 , ,_ 2(g-l) _ 6 2 
12’ Ti('^) 12 “12’ 'tu'^) 12 
I ^ N (s-l)Hs^ 25 
4,^ = f2 =I2 
i'/,N 2(^-l) + 2g 14 
'Y2(v— 12 “12’ '^2w)“12 
x^ + 3<^’" 
' liT 
, (s — 1)^ + 5^+ (s+ 1)^ 50 
Ysi*’)— 12 “12’ 
Vf \ — l) + 2s + 2(s + l) 
Y3\^) — J2 
24 
12 ’ 
12 
'FW=i|o.4,+9.4,,,+25.4..,+S0.4.^,j 
T'(s)=^{o.4,+6.4,-,+ 14.4,_,+24.4,_,} 
Y'(s)=^{o.4,+2.4,_,+4.4._,+6.4,^,J, 
( 60 .) 
