About the Book
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. 1827 Excerpt: ...we put x2 =y, the following form: /--Ay3 + Bf--Cy + D = o and the roots of this equation are (Vp + Vq + Vr)"; (Vp + Vq-Vrf (Vp-Vq + Vr)2, (Vp-Vq-Vrf 3. In order to determine from hence the coefficients A, B, C, D, wc only need take the sum of these roots, the sum of every two of them, and so on. The following treatment, which has been frequently made use of already in the preceding part of this work, leads to the object in a shorter way. Let S1, S2, S3, S4, denote the sum of these roots, the sum of their squares, cubes, and fourth powers; then, when in IX, --A and C are put for A and C, and the symbol S for the one there used, A = S1 B-ASl-Si 2 c _ BSl-ASz + S3 3 D = CS--B + S3--S4, 4 Solution 1. By the two preceding it is easily inferred, that when n is the number of the irrational magnitudes Vp, /q, ... Vie, the degree of the rational equation is equal to the power 2." But since the different values of x are such, that two of them are always similar, but with different signs, the equation consequently is only of the 2"-1th degree, when we put x = y. 2. The conclusions in the two preceding, when ex tended, give the following results: 3. Hence the law of the formation is easily perceived. As an example, I will take SS. The number 5, and its divisions into combinations of two, three, &c, give the numerical expressions 5, 14, 23, 13, 122, 12, l5. The coefficients are no other than the number of transpositions of different things, whose repeating exponents are twice as great as the radical exponents of the numerical expression; consequently the coefficients of 5, 14, 23, l23, 122, '132, l5, the number of transpositions of the different things a10, a268, a466, aW, aW, .a%W, a2WcPe2, or 1, 45, 210, 1260, 3150, 189...
