398 MAJOR P. A. MACMAHON ON THE THEORY OF PARTITION’S OF NUMBERS. 
in succession, and we may combine any number of such conditions as are inde¬ 
pendent. 
Art. 122. I postpone further investigation into this interesting theory, and will 
now give a formal proof tha.t the product tableau is, in fact, finite and integral. The 
product in question for > Hs 
XjX: . 
.. x;:}{x,x,,, 
. . . 
11 
for 
Al 
11 
for 
s ^ 1 and < on 
+ 1 
= 1 — S 
for 
i 
IV 
All the conditions may be resumed in the single formula 
a. 
+ + i + “3S+2 
'.t + • 
. . > 
+ ^ 2 , 
s + 2 ( ~h “3.s + 3t 
+ . . . 
s and t being any 
integers. 
Let the 
greatest 
■ , . 1 t — . 
integer in 
^ s + ^ 
be denoted 
'vC:; 
- 1 
byl 
[ simply for 
brevity. 
Similarly let I 2 refer 
on 
to — 
s 
±t 
+ t ’ 
I 3 to 
1 + 
m + t — I 
s + ^ 
J 1 to 
s 
- I 
Jto , 
s + C 
and J 3 to 
1 -f i n — 
We derive 
-S "f- 
ll 
- Ji 
or 
Ji 
-b 1 
I 2 
= J.3 
or 
J 2 
+ 1 
I 3 
= J 3 
or 
Ja 
+ 1 
Ii 
+ I 2 
= l3 
or 
I 3 
+ 1 
Ji 
+ J 2 
= Ja 
or 
Ja 
— 1 
and we have ten possible cases to consider, viz. :— 
Case 1. 
L = J. 
li 
+ J2 — ts 
1.2 =■ J.. 
T _ T 
Ji 
“b J2 — J3 
^3 - ^3 
Case 2. 
b = Ji + 1 
It 
H- I2 = la 
L = J2 
Ji 
~b J 2 J 3 — 1 
I3 - Ja 
Case 3. 
b = Ji + 1 
L 
“b L = I3 -b 1 
L = J2 
Ji 
-f- Jo — J 3 
I3-J3 
