Solutions of the Cambridge Problems Volume 1; From 1800 to 1820

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1825 Excerpt: ...--5 = 0. By trial, X--2.1 nearly. Let X = 2.1 + v Thenx 9.261 +(4.41) X --2x---(4. 2)--2- Warly = 11.23i +.061=0-5 3-5 J v=.--'G =--.00543 nearly. 11.23 X == 2.09457 nearly. With this new value of x repeat the operation, and it will be found that x more nearly = 2.0945515 For the required explanation, see the Introduction to Barlow's Mathematical Tables, page 31, where the defects of the various methods of Approximation are clearly pointed out. This valuable book should be in the hands of every mathematician. Then it is known that P = r + 2r + 3r +.... (n-l)r--366. Besides the m real roots, there can only be pairs of imaginary roots of the form a b J--1, where, a is positive negative or zero, (b is necessarily unequal to zero). Now, the differences arising from combining imaginary roots of the form a --1 or of 6--1, b'tj--1, &c, are of the form (b + b') J--1, whose square is negative. All other differences in which two imaginary roots, or in which an imaginary, and a real root are involved, are of the form a b J--1, the square of which is imaginary. Again, since there are m real roots, taking them two and two we form m.--L differences, the squares of which are positive..'. with regard to sign, m.--I of the roots of the trans formed equation are positive, and the rest either negative or imaginary. Hence, the last term being = product of all the roots with their signs changed, can only be affected in sign by the . m 1 real 2 fit 1 roots. It is positive or negative, according as m.--_--is even or odd. 367. This can be effected by the Method of Divisors, since there must be corresponding rational factors equal to these roots in the last term; unless any of the ...