Elements of Algebra

320 ELEMENTS PART II. example, let $m = 12$, $a = 5$, $b = 7$, and $c = 15$, and we have, for $my-a = \frac{mc+ab}{mx-b}$, $12y-5 = \frac{215}{12x-7}$. Now, the divisors of 215 are 1, 5, 43, 215; and we must select from these such as are contained in the formula $12x-7$; or such as, by adding 7 to them, the sum may be divisible by 12; but 5 is the only divisor that satisfies this condition; so that $12x-7=5$, and $12y-5=43$. In the same manner, as the first of these equations gives $x=1$, we also find $y$, in integer numbers, from the other, namely, $y=4$. This property is of the greatest importance with regard to the theory of numbers, and therefore deserves particular attention. 36. Let us now consider also an equation of this kind, $xy + x^2 = 2x + 3y + 29$. First, it gives us $y = \frac{2x-x^2+29}{x-3}$, or, by division, $y = -x-1+\frac{26}{x-3}$; and $y+x+1 = \frac{26}{x-3}$; so that $x-3$ must be a divisor of 26; and, in this case, the divisors of 26 being 1, 2, 13, 26, we obtain the three following answers: 1. $x-3=1$, or $x=4$; so that $y+x+1 = y+5 = 26$, and $y=21$; 2. $x-3=2$, or $x=5$; so that $y+x+1 = y+6 = 13$, and $y=7$; 3. $x-3=13$, or $x=16$; so that, if $y+x+1 = y+17 = 2$, $y$ must be $=-15$. This last value, being negative, must be omitted; and, for the same reason, we cannot include the last case, $x-3=26$. 37. It would be unnecessary to analyse any more of these formulæ, in which we find only the first power of $y$, and higher powers of $x$; for these cases occur but seldom, and, besides, they may always be resolved by the method which we have explained. But when $y$ also is raised to the second power, or to a degree still higher, and we wish to determine its value by the above rules, we obtain radical signs, which contain the second, or higher powers of $x$; and it is then necessary to find such values of $x$, as will destroy the radical signs, or the irrationality. Now, the great art of Indeterminate Analysis consists in rendering those surd, or incommensurable formulæ rational: the methods of performing which will be explained in the following chapters*. * See the Appendix to this chapter, at Art. 4, of the Additions by De la Grange.