500 RESOLUTION or IN jpLTERM IN ATE PROBLEMS.
By proceeding, therefore, as in the aforegoing cases,
we have x — 8331 — y— * ; whence the least va-
lue of y is given n 3, and the greatest of x — 6328.
Moreover, from the equation 4 m f 5n - 20, we have
yi zz b— n —whence n zz 0, and m — 5.
Therefore the general values of x and y [gken in Problem
13) do here become 8328 — 5q — 5u, and 3 4q;
from the former of which the greatest limit of q is given
8328
= -—-= 1665. Now, since the value of y will here
continue positive, in all substitutions for q and u (as
no negative quantity enters therein); the whole num
ber of answers will be determined by the values of x
alone.
In order to this, let q be successively expounded by
1665, 1664, 1663, &c,
and it will thence appear
that the said number will
be truly defined by 1666
terms of the arithmetical
progression \ f 2 f 3
-f 4 +- 5, &c. whereof
the sum is found to be
J38S611»
7
X
Quot.
N.Ans.
1665 3—bU
1664 8—5 U
1663 13—5U
&C. j &C.
Of
If
2|
&c.
1
2
3
&C.
When there are four indeterminate quantities in the
given equation, the number of all the answers may be
determined by the same methods: for, any one of those
quantities may be interpreted by all the integers, suc
cessively, up to its greatest limit (which is easily de
termined); and the number of answers, corresponding
to each of these interpretations may be found, as above;
the aggregate of all which will consequently be the
whole number of ans\yers required: which sum, or
aggregate may, in many cases, be derived by the me
thods given in Section 14,* for summing of series’s by
means of a known relation of their terms. But. this
being a matter of more speculation than real use, 1 shall
now pass on to other subjects.