MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 38-3 
Conditions. 
Result. 
a. 
''><7 ^ a4 
«■>, 
'M 
a. 
+ 
,.ii 
+ 
(1)(2) (3) (4) (5) (6) (7) (8) 
a- >- ag > aj 
^5 ^ a-j «5 — «.i 
> ag > «4 
«2 > a., a., > ag 
cc- >. ag > 
cig > a.,, ag > ag 
as' + ,d + 
(I) (2) (3) (4) (5) (6) (7) (8) 
_■>•' F _ 
(1) (3) (3) (4) (5) (6) (7) (8) 
aA + a;' + a;^" 
(1)(2) (3) (4) (5) (6) (7) (8) 
and by addition the resulting generating function'"" is 
1 4- 2./;- + 2a"' 4- 3./4 4- 3a-’ 4- oa® 4- 4a' 4- 8a/ 4- + oa^" 4- -y^ + 3ai' 4- 2a‘" 4- 2aj‘‘ 4- a' 
(1) (2) (3) (4) (5) (6) (7) (8) 
Art. 99. By analogy with the lattice in piano one might have conjectured that the 
result would have been 
1 
(1)(2)R3)R4)’ 
but this is not so, although the twm functions do coincide as far as the coefficient of 
inclusive. In fact, the two expansions yield respectively 
1 -|“ ^ "b 4a;‘ -}“ lx’ “b 14.r^ -b '23x^ -b 41.a/ -j~ G3.r' -b . . . , 
1 + a: + 4a;- + 7.// + 14.r' + 2.3 .'b“ + 42.x'" + 63x‘ + . . . , 
the succeeding coefficients becoming widely divergent. This at tirst seemed sur¬ 
prising, but observe that analogy might also lead us to expect that, if the part- 
magnitude be limited to /, the result would be 
(i 4- I) (t 4- 2)'(< 4- 3)-’ (7 4" 4) 
(1) (2f‘ (3)^ (4) ’ 
but this does not happen to be expressible in a finite integral form for ail values of /, 
a fact which necessitates the immediate rejection of the conjecture. The expression 
in question is only finite and integral when i is of the form Sp or 3p + 1. We have. 
* Mr. A. B. Kejite, Treas. R.S., has verified this conclusiou by a dificrent and most iugenious mctliod 
of summation, which .also readily yields the result for any desired restriction on the part-magnitude. 
