be found, so that y may be an integer ; and, in the fourth, x is to be found, fo that y may be a rational quantity. 1. When y=4+6*, the folution depends on the rule for deducing a fraction to its lowest terms. See the text, page 125. 2. When y = it is plain that; in order to have y an integer, b+cx must be a divisor of a. Let d be one of its divifors, then, if b+cx=d, x=; b; so that among the divisors of a, we must find one, if possible, from which b being taken away, the remainder may be divisible by c: the quotient is a value of x. if d be a divisor of 3. When y=a+ bx +dx' b, x will be taken out of the numerator, if we divide it by dxtc; and this form is then reduced to the preceding. But, if d is not a divisor of b, multiply both sides by , or, dividing bdx tad by d', then dy=adtbdx by dx+c, dy=b+adambi , and so x is found cfdx Example. -1+ 391 Let 2xy+x+y=195; then y (2x+1)= and 195—* . Therefore 2y = 1+2x 390-2x Now 391=17X23; 1+ 2x 4. When y=va+bx+cx?, and x is to quantity. Here there are four cases, according to the nature of the coefficients a, b, and c. Imo, If a be a square number, as for instance g?, so that the formula is vg2+bx+cx?. Suppose Vg2+bx+cx2 = 8 + mx, then g2+bx+cx?=g2+2mgx+m2x, or bx+ cx2 2mgx + m2x2, that is, b+cx = 2mg +m’x, and x= 2mg-6 cm2 If appear, and If for x this value of it be substituted in the given formula, its irrationality will dis and vg2+bx+cxt = cg—bm+gma m may be assumed, therefore, equal to any quantity, positive or negative, integral or fractional, and the corresponding value of it will answer the conditions prescribed. 2do, If c be a square number, as g?, then let Va+bx+g2x2=m+gx. Hence a+bx+g2x2=m2 +2mgx +g4x?, or a+bx ma =m2 +2mgx. Therefore x = and ----2mg Va+bx+go2x2 Here my as before, may be assumed at pleasure. 3tio, Though neither a nor c are square numbers, yet if a+bx+cava can be resolved into two simple factors, as f+gx, and b+kx, the irrationality of the formula may be taken away. For, let Va+bx+cx2 v (f+gx) (b+kx)=m(f+gx), and (f+gx) (b+kx) =m2(f+8x), or h+kx = m2 (f+ex), and x = By the sub k--gm fiitution of this value of x, (m being afsti med b bm-gm-ag. b2mg 2 med at pleasure), the irrationality will be removed, as before. 4to, The fourth case, in which the formùla a+bx+cx? may be rendered a complete square, is when it can be divided into two parts, one of which is a complete square, and the other a produd of two simple factors. For atbx+cx? is then of the form pa-qr, , q, and r, being quantities into which there enters no power of x higher than the first : And, if we assume ✓patar=p+mq, p2 will be exterminated, and the remainder, being divided by 9, will be a simple equation, from which x may be easily determined. These methods of removing the irrationality of the preceding formula are to be particularly attended to, as being of great use in the higher geometry. APPEN |