Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

THE STUDY OF THE PROPERTIES OF CONSECUTIVE INTEGERS. 17
The fact is that we can reduce the three relations (a), (b), (c) to two by writing a2 + a
for a2, a tranformation that the relation (b) permits, and then we have to write N2NVs for
N23 in the sum
We next eliminate c, obtaining
Q 1
i1
> 1-a aN. 1-1- N2. 1 - a2NlN2lAT3
a
an expression that would have presented itself if we had been summing
YN' iN22a (N N2T33)a3
for the single relation a, + 2a3 > a2,
obtained from the relation
al + aO3 a2,
by writing a + a3 for al, a transformation permitted by the relation
al > a3.
The process employed is therefore equivalent to a gradual reduction in the number
of the conditional relations associated with a proper transformation of the product to be
summed.
To eliminate a we require the subsidiary theorem
IQ i__ 1 ___ 1 + 1 + xy- xyz - xyz2
i 1 - x. 1 - y. 1 - Y.yz. 1 - xz2
1 - a2x. 1- ay. I -z
a
and thence we derive
1i + N1N22N3 - N12N2T3 - N1N23n,
1 - N. 1 - NN,2. 1i - N1N2N3. 1 - NlV23N3
i - N2N1Y - N2N23N 1 - N12N24AT32 + Nl3N24V 39 + N1325N32
- N.1i N. 1 - N. 1 -NN - 1-NNiV 3.1 - 1-N1323NT3
In this result the denominator indicates the ground products, and the numerator the
simple and compound syzygies which connect them.
It is manifest that the ground products are
N-, NN2,, N-N2N3, N1AT22N3, N1AT231T3
connected by the simple syzygies
(A) = (NV) (N1N22N3) - (TÇ17T2) (N1N2N3) = 0,
(B) = (N1) (N1N2N33) - (VN2,) (NV,2N3) = O,
(C) = (NN,2N) (NN2,3N3)- (NN22N3)2' = o;
VOL. XVIII. 3