Lessons Introductory to the Modern Higher Algebra (Classic Reprint)

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Excerpt from Lessons Introductory to the Modern Higher AlgebraIt may be said to date from a paper published by Professor Boole in the Cambridge Mathematical J our nal for November, 1841, which was mainly occupied with applications of the following theorem: Let an or dinary algebraic equation be made homogeneous by writing a: y for 12, let these variables then be linearly transformed by writing M: pg, Xx My for x and y and let V 0 be the condition that the transformed equation shall have equal roots, then the function V will be equal to the similar function for the original equa tion multiplied by a function of A, p, N, pf, and not in volving the coefficients of the original equation. The truth of this principle is manifest from the fact that if the original equation have a square factor, the trans formed must have one likewise, that, therefore, one of the functions in question must vanish whenever the other does, and, consequently, the one must contain the other as a factor, and since each of the coefficients of the transformed only contains in the first degree the coefficients of the original, the two functions, being of the same degree in those coefficients, can only differ by a quantity independent of them. Similar reasoning ap plies to functions of any number of variables.About the PublisherForgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.comThis book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully, any imperfections that remain are intentionally left to preserve the state of such historical works.