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. 29
and observing that we may put b=f = 1 and that moreover
xz = (12111), yw = (131),
py2 = (121) (131), pxy = (121) (14211),
px2 = (13211)(14211), pxw = (13211)(14211)2,
py3z= (121)3, xw = (14211), yz = (111),
while operation upon the remaining letter c produces no new form, it is clear that no new
form arises.
Case 4. a2>, a 2 > a, a3, > a + a5.
The reduced conditions are
a2 > a,................................................. (a ),
a3. >...........................................(b),
3 > ai + a4.............................................(c),
2al + 2a 5 ) 1 + 2a4............................................ (d),
ai + a4 > a3..................................................(e),
a, 3 a 04.................................................. (/ ),
o........... (ye),
a > a5.........................................( ),
a >..................................................( );
leading to NV1 12Ta,3N3Na,-/4Nj5
1
IIn ______ ac
= - d2eh la JT 1 bf c dN
a1 N1.1 d-b 2 1 _ e3 1 e d2 N'
ac d be d-f 4cgh
and this by elimination of a, b, c, e, g and h becomes
n d N2N3N4
Q 1
d 1 -^/(lll). i 1 1 - d2f. 1 - 1 5N, N,
and since > 1
1 -fx. 1 -fy. 1 -f w
1 + yw
1-. 1-y. 1-Z.1-zw 1-y. 1-yw. 1-zw
xw x 2w2 _
1- x.1- y.1 - yw. 1 - zw _1- x. 1- xw. - yw. 1 - zw