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. 15
so that the solution of a, > fil
Q
is given by
- mx,. 1
Again, if a - ii + 2,
f 1
we have
- -mr.-n.-1 1
-m-x,. 1 - - Yi. 1 Y2- -
QZ 1
mm
1 - xm. 1 - x yl. I - xy
the solution.
Also the solution of a, > fi + fi2
is XI
S -1 x. 1 - xy. 1 - x-Y2
Lastly, ai + a2 > f
gives, by repeated application of the above simple theorem,
1 - xix2yi
i - x -. - 1. - xyl. 1 - xy
In general the subsequent work merely involves processes easily derivable from these cases.
Particular theorems will be given as they become necessary, and for the general theory,
which is here not needed, the reader is referred to Part III. of the Memoir on Partitions
which may appear shortly in Phil. Trans. R.S.
To come to the object of the paper I commence with
ORDER 2.
(n+ 1 )Cn+2 )a2
this product is an integer when n is even, but when n is uneven we must have
ai > a,;
and $NalNa2= Q i
a
1- _,. i- - N,'