Diminish each constituent of the first column by twice the corresponding constituent in the second column, and each constituent of the fourth column by three times the corresponding constituent in the second column, and we obtain and since the second row has three zero constituents this determinant 505. The following examples shew artifices which are occasionally useful. Example 1. Prove that a b с d = (a+b+c+d) (a − b + c − d) (a − b − c + d) (a + b − c −d). b a d с с d a d с b a By adding together all the rows we see that a+b+c+d is a factor of the determinant; by adding together the first and third rows and subtracting from the result the sum of the second and fourth rows we see that a-b+c-d is also a factor; similarly it can be shewn that a-b-c+d and a+b-c-d are factors; the remaining factor is numerical, and, from a comparison of the terms involving a on each side, is easily seen to be unity; hence we have the required result. Example 2. Prove that 1 1 1 1 (ab) (ac) (ad) (bc) (b-d) (c–d). = a3 b3 c3 d3 The given determinant vanishes when ba, for then the first and second columns are identical; hence a − b is a factor of the determinant [Art. 514]. Similarly each of the expressions a-c, a-d, b-c, b-d, cd is a factor of the determinant; the determinant being of six dimensions, the remaining factor must be numerical; and, from a comparison of the terms involving bc2d3 on each side, it is easily seen to be unity; hence we obtain the require result. 480 HIGHER ALGEBRA. CUBIC EQUATIONS. 575. The general type of a cubic equation is x2 + Px2 + Qx + R = 0, but as explained in Art. 573 this equation can be reduced to the simpler form x3+qx + r = 0, which we shall take as the standard form of a cubic equation. 576. To solve the equation a3 + qx+r = 0. Let x = y +z; then x2=y3 + z3+3yz (y + z) = y3 + z3 + 3yzx, and the given equation becomes y3 + z3 + (3yz + q) x + r = 0. At present y, z are any two quantities subject to the condition that their sum is equal to one of the roots of the given equation; if we further suppose that they satisfy the equation 3yz+q=0, they are completely determinate. We thus obtain hence y3, 23 are the roots of the quadratic we obtain the value of x from the relation x = y +z; thus (1), (2), The above solution is generally known as Cardan's Solution, as it was first published by him in the Ars Magna, in 1545. Cardan obtained the solution from Tartaglia; but the solution of the cubic seems to have been due originally to Scipio Ferreo, about MISCELLANEOUS EXAMPLES. 515 231. It is known that at noon at a certain place the sun is hidden by clouds on an average two days out of every three: find the chance that at noon on at least four out of five specified future days the sun will be shining. [QUEEN'S COLL. Ox.] 233. Eliminate a, y, z from the equations: x2-xy- -XZ y2-yz-yx _ 22-zx-2y, and ax+by+cz=0. α b с [MATH. TRIPOS.] [QUEENS' COLL. CAMB.] 234. If two roots of the equation a3+px2+qx+r=0 be equal and of opposite signs, shew that pq=r. 236. If (1+a3x1) (1+a3ñ3) (1+a3x16) (1+a17x-32)................ ...... =1+A4x2+Agx8+A12¤12 + prove that Agn+4a3Agn, and Agn=a2 An; and find the first ten terms of the expansion. [CORPUS COLL. CAMB.] 8n 237. On a sheet of water there is no current from A to B but a current from B to C; a man rows down stream from A to C in 3 hours, and up stream from C to A in 3 hours; had there been the same current all the way as from B to C, his journey down stream would have occupied 2 hours; find the length of time his return journey would have taken under the same circumstances. 238. Prove that the nth convergent to the continued fraction 3 3 3 2+ 2+ 2+ is 3n+1+3(-1)+1 [EMM. COLL. CAMB.] 239. If all the coefficients in the equation x2+P1x-1+P1an-2+...+Pn=f(x)=0, be whole numbers, and if ƒ (0) and ƒ (1) be each odd integers, prove that the equation cannot have a commensurable root. [LONDON UNIVERSITY.] SOLUTIONS OF THE EXAMPLES IN HIGHER ALGEBRA BY H. S. HALL, M.A. FORMERLY SCHOLAR OF CHRIST'S COLLEGE, CAMBRIDGE AND S. R. KNIGHT, B.A. FORMERLY SCHOLAR OF TRINITY COLLEGE, CAMBRIDGE London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY PRESS OPINIONS Academy "A fitting complement of the text-book. . . . We have read a great number of these solutions, and have found them well suited to their purpose, as they are concisely put, and yet no necessary steps are suppressed." Schoolmaster-"This Key to the authors' Higher Algebra is a companion in the best sense of the word. It not only contains solutions to nearly all the questions in the original work, but includes valuable hints as to processes, and judiciously guides the student wherever a word or two will help him in his study, and save time and labour in the elucidation of knotty points. . . . Such a Key will be of great use to teachers generally, and specially serviceable to that large class well known as 'private students.' We can heartily recommend the volume as educational and practical in an eminent degree." Nature-"The volume will prove most useful to teachers, and we strongly recommend it to students who are beginning the study of Algebra without a teacher." H. S. HALL, M.A., and S. R. KNIGHT, B.A. THIRD EDITION, REVISED AND ENLARGED Saturday Review-" To the exercises, one hundred and twenty in number, are added a large selection of examination papers set at the principal exam. inations, which require a knowledge of algebra. These papers are intended chiefly as an aid to teachers, who no doubt will find them useful as a criterion of the amount of proficiency to which they must work up their pupils before they can send them in to the several examinations with any certainty of success. Irish Teachers' Journay.-"We know of no better work to place in the hands of junior teachers, monitors, and senior pupils. Any person who works carefully and steadily through this book could not possibly fail in an examination of Elementary Algebra. We congratulate the authors on the skill displayed in the selections of examples." Schoolmaster-"We can strongly recommend the volume to teachers seeking a well-arranged series of tests in algebra." PREFACE THIS book consists of one hundred and twenty progressive miscellaneous Exercises, followed by a collection of Papers set at recent Examinations. The EXERCISES have been frequently tested among our own pupils, and each will be found of suitable length for about an hour's work. They are arranged as follows: Part I. takes in the early rules up to, and inclusive of, Simple Equations; in Part II., Involution, Evolution, and Simple Fractions are introduced; Part III. takes in Resolution into Factors, and Fractions of all kinds; Part IV., Quadratics, Indices, and Surds; Part V., Ratio and the Progressions; Part VI., Permutations, Combinations, and the Binomial Theorem; and Part VII. consists of Papers of Miscellaneous Equations of the most useful types. The EXAMINATION PAPERS are more varied in length and character; they will be found to comprise specimens of papers set at all the most important examinations in which a knowledge of Elementary Algebra is required. |