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. 33
and
Q ___________________
1 - x.. 1 - c2. 1 - z
i z
- +
1-x.l-y.1 -w 1-x.-y. 1 -w. 1-wz
XZ2 Z2
-+ ___ +yz
1-x.1-y.l- zz. 1-yz 1- y. 1-wz. 1-yz
Xz3 + X2Z4 + 3xz5
1-. 1 -wz. 1yz. 1- x '
Putting now
x=(11211), y=(111), w= (12211), z=(0111),
we can examine the generating function.
It is clear that
xz= (12321)
is a ground product.
Also x2z = (12321) (13431) = (25752)
is a ground product, (13431) not being a solution of the conditions.
Further
(12211) (12321)z = (12321) (13321),
wz = (13321),
(12211) (12321) yz2 = (121) (25752),
(12211) (12321) xz3 = (13321) (25752);
so that there are no more ground products.
We have therefore in Case 6 obtained the new fundamental forms:
(12321),
(25752).
The investigation that has been given does not establish that the 13 forms obtained
are ground products quz the whole of the six cases, but it does prove that all the ground
products are included amongst these 13. But it is clear that all forms in which ac=l are
necessarily ground products. This accounts for 9 of the 13 and it is easy by actual experiment to convince oneself that the remaining 4, viz.:(25752),
(32121),
(33232),
(34343),
are, in fact, irreducible.
VOL. XVIII. 5