SUPPL

SUPPL. VOL. II. Part I.

with the other parts of algebra. The solutions commonly given are devoid of uniformity, and often require a variety of assumptions. The object of this paper is to resolve the complicated expressions which we obtain in the solution of indeterminate problems, into simple equations, and to do so, without framing a number of assumptions, by help of a single principle, which, though extremely simple, admits of a very extensive application.

"Let $A\; \backslash times\; B$ be any compound quantity equal to another, $C\; \backslash times\; D$, and let $m$ be any rational number assumed at pleasure; it is manifest that, taking equimultiples, $A\; \backslash times\; m\; B\; =\; C\; \backslash times\; m\; D$. If, therefore, we suppose that $A\; =\; mD$, it must follow that $mB\; =\; C$, or

$B\; =\; \backslash frac\{C\}\{m\}$. Thus two equations of a lower dimension are obtained. If these be capable of farther decomposition, we may assume the multiples $n$ and $p$, and form four equations still more simple. By the repeated application of this principle, an higher equation, admitting of divisors, will be resolved into those of the first order, the number of which will be one greater than that of the multiples assumed."

For example, resuming the problem at first given, viz. to find two rational numbers, the difference of the squares of which shall be a given number. Let the given number be the product of $a$ and $b$; then by hypothesis, $x^2\; -\; y^2\; =\; ab$; but these compound quantities admit of an easy resolution, for $x\; +\; y\; \backslash times\; x\; -\; y\; =\; a\; \backslash times\; b$. If, therefore, we suppose $x\; +\; y\; =\; ma$, we shall obtain $x\; -\; y\; =\; \backslash frac\{b\}\{m\}$; where $m$ is arbitrary, and if rational, $x$ and $y$ must also be rational. Hence the resolution of these two equations gives the values of $x$ and $y$, the numbers sought, in terms of $m$; viz.