MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 397 
indicating the ground forms 
Y,Y.,Y3Y, = X.X,X3X,X,X,X,X, 
Y1Y3YIY! = XiX,XlXiX?Xi:X,X8 
Y,Y;;YiY' = X,X^X^X!XrXjXrX, 
YiY^YiYf = XiXiX^X^X^X^XrXs 
YiY^Y^Y! = X.X^X^X^X^XgXrX^ 
V V-V^V^ — Y" Y-Y^Y^Y-iY3Y-Y 
Art. 121 . So far it appears that all products which can he placed in the form of a 
rectangle 
X: X, X 3 ... X, 
X,X3 X4 ...X,,, 
X,iiX,jj_)_ jX.,)j^.2 . . . X.;^.,JJ_1 
are ground forms for all values of / and m. 
I have established this independently, and thus proved that the conjectured 
result for the general lattice in piano is, at any rate, linite and integral, as it 
should be. 
It is desirable to ol^tain information concerning the ground forms which are not 
within the rectangular tableau. 
The forms 
XpX?=. . . Xy, 
which appear in the tableau, may be eliminated from consideration, with the 
exception of the form 
X,X2... x„ 
by ascribing additional conditions such as 
ai = a,, 
which are not true in the tableau. 
The condition of this tableau is that if a^, = “p+n no index is greater than a.^ ; 
after a repetition of index, no rise in index takes place, in the Y form, therefore, 
we may assign the conditions 
U.p ^p+\ + l 
for any value of j;, as one excluding the whole of the forms a])pertalning to the 
tableau. 
We may impress the conditions 
aj =; a,, < ag 
a., = aj -=: 
a = < 'a-^ 
