NUMBER OF PARTITIONS OF WHICH A GIVEN NUMBER IS SUSCEPTIBLE. 417 
12 
'Tii 
12 ^+ 8 . 12 ,_.- 1 . 12 ,_,+ 8 . 12 ,_ 3 + 2 . 12 ,_ 4 + 4 . 12 ,_ 5 - 2 . 12 ,_ 6 + 8 . 12,_7 + 
+ 1 . J 2,_,+0. 12,„5+ 1 . 13,_..+4 . 12,_„|. 
And assembling these several portions, Xd- Y+(Z'+Z"+Z"'), we get 
n(j:)=j^{a^*+3j?^-9a;.2,_,}+Y^{0.12,+5.12,_,-20.12,_3-27.12,_3+32.12,_4 
-11.12,_3-36.12,_6H-5.12,_7+16.12,_3-27.12,_9-4.12,_.o-11.12,_„j. 
(37.) Proceeding now to the case where s=5, we have 
+ ,, + 6a; + 6 ,,, 6 
?’(*)=-i5^. f W=^ir> p W=n4’ 
whence 
^ (x + 4}^(a; + 7) 3(^ + 4)(a;’ + 6) 6(^ + 5) ^ 6 
^(j^ + 4)= ^44 <p(^ + 4)=^ f{x+A) = -Y^, 9"(^+4) = ^, 
and executing the reductions, arising from the substitution of these in equation (35.), 
we find 
«4 + 10^ + 19a^2_22a; 
x=- 
2880 
Again for Y we have 
'4'i(5) = ‘P(4) ; 4'2(5) = 9(4) + <P(5) ; '4^3(S) = <P(4) + <?>(5) +?’(6)j &c., 
whence 
' 4 'i(^) — 144 ’ r 2 ( 3 ) — 144 ’ 'Yai^J — 144 ’ — 
4i(5)=fis; 4U5)= 
144 
144 
510 
144 
'4'i(S) — 144’ — 144’ 144’ 144 
^:(5) = ^; ^:(5)=^^; ^7(5) — 
■^^(5)=:^4{o.5,+ 112.5,_, + 312.5,_,+636.5,_3+1126.5,_4} 
T(5)=^|o. 5^+72. 5._, + 177 . 5._.H-321.5._3+510.5,_,| 
T'(5)=:y^^|o.5,+30.5._i + 66.5,_2+108.5,_3+156.5._4} 
■^"'(5)=i^{o.5,+6.5,_, + 12.5,_,+ 18.5,_3H-24.5,_4} 
= -^{o.5.+8.5,_,+ 128.5,_3+456.5,_3+1112-5,_4}. 
3 H 
MDCCCL. 
