MAJOR P. A. MACMAHON ON THE THEORY OF PARTITIONS OF NUMBERS. 367 
the ground solutions are 
(«!, a,, ^3, «4) = (0, 1, 0, 0), (1, 1, 0, 0), (0, 1, 1, 0), (1, 1, 1, 0), (0, 1, 1, 1), (1, 1, 1, 1) ; 
the three simple syzjgies are given by 
Xo.XiX,X3 - X1X2.X0X3 = Si = 0, 
X2.XiX,X3X, - XjX^.X,X3X4 = S3 = 0, 
X^Xa.XiX.^XsXi - XiX3X3.X3X3X4 = S 3 = 0, 
and the two compound syzygies by 
X-^XsXi.Si 
X 1 X 3 X 3 .S 3 
X 2 X 3 .S, 
XiX^.Ss 
0 , 
0 . 
Art. 80. Ill general, when the number of parts is we have k\ orders Avhich are 
altered by interchange of d and a, combined with inversion, and I, which are un¬ 
altered where 
2 ki -|- 4 = 
Hence the number of essentially different orders is 
hi li = ^l.,. 
To determine observe that an order 
d-^-a'^^ . . . d^''‘a'''"^ d^‘a"' 
will be unaltered by the operations spoken of when 
K — l-^s — H-i — K = K — p-s-i = H-2 — K-i = . . . = 0 ; 
so that i — 1 must be even and there will be two such unaltered orders for each 
partition of . i — l .into even parts. 
Hence the generating function for -j- k is 
X- 
giving for 
1 — 2x (1 — v) (1— .j;‘) (1 — ?/’). . . rrcl ivf 
i = 2, 3, 4, 5, 6, 7, . . . 
1, 3, 4, 10, 16, 35, . . 
the numbers 
