Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1876 edition. Excerpt: ... The impossibility of the reduction indicates some singularity in the form of the quantic. Thus a cubic having a square factor cannot be brought to the form Ax' + Dy" a different canonical form must be adopted, and the most simple one is the form xy, to which the cubic in question is obviously at once reducible. 168. In the same manner as a cubic can be brought to the sum of two cubes, so in general any binary quantic of odd degree (2n-1) can be reduced to the sum of n powers of the (2n-l)m degree, a theorem due to Dr. Sylvester. For the number of constants in any binary quantic is always one more than its degree, or, in the present case, 2n; and we have the same number of constants if we take n terms of the form (lx + myY"'1. The actual transformation is performed by a method which is the generalization of that employed (Art. 166). For simplicity, we only apply it to the fifth degree, but the method is general. The problem then is to determine u, v, w, so that (a, 5, c, d, e, f$x, yT may = w6 + + w5. Now we say that if we form the determinant ax + by, bx + cy, cx + dy bx + cy, cx + dy, dx + ey cx + dy, dx+ ey, ex +fy, In general, when a binary quantic has a square factor, this will also be a square factor in its Hessian, as may be verified at once by forming the Hessian of x-j. uvw lm'-I'm' I'm"-l"m'f (l"m-lm") When, then, the determinant written in the beginning of this Article has been found, by solving a cubic equation, to be the product of the factors (x 4 y) (x 4 fty) (x + vy), we know that u, v, w can only differ from these by numerical coefficients, and we may put a, I, c, d, e, fjx, y) = A(x + yY + B(x + fiyf +Cx + vy)6; and then A, B, C are found by solving any of the systems of simple equations...
