TRANSFORMATION of equations. The doctrine 'of the transformation of .equa tions, and of exterminating their inter mediate terms, isthustaught by Mr. Mac Laurin. The affirmative roots of an .equa tion are changed into negative, roots of the same value, and. the negative roots into affirmative, by only changing the signs of alternately, with the second. Thus the roots of the vert t- x3— 19 EC' + — 30 =0, are +1, + 2, + 3, —5; whereas the roots of the same equation, having only the signs of the second and fourth terms changed, vi:, x =0, .
To understand the reason of this tale, let us assume an equation, as x—a x x—b X s—c X x—e, 0, whose roots are + a, ± b, c, d, + e, &c. : and another, having its roots of the same value, contrary signs, as x+a ,X x Xx c X X x e, R &c. = 0. It is plain, that the terms taken alternately, beginning from the first, are the same in both equations, and have the 'same sign, being products of an even num ber of the roots ; the product of any two roots having the same sign as their pro duct when both their signs are changed; as-FaX—b.---ax+ b.
But the second terms, and all, taken ternately from them, because their cients involve always the products of an odd number of the roots, will have trary signs in the two equations. For ample : the product of four, via. a b c d, having the same sign in both, and one equation in the fifth term having a b c d e, and the other a b c d X— e, it follows, that their product, a b c d e, must have contrary signs in the two equations: these two equations, therefore, that have the same roots, but with contrary signs, have nothing different but the signs of the alternate terms, beginning with the second. From which it follows, that if any equation is given, and you change the signs of the alternate terms, beginning with the second, the new equation will have roots of the same value, but with contrary signs.
TRANSiT, in astronomy, signifies the passage of any planet, just by, or over fixed star, or the sun, and of the moon in particular, covering or moving over any planet. Sre