Academy-"These solutions are just what are wanted by mathematical masters who cannot command the time necessary for the immediate working-out of many of the exercises whilst engaged with their classes. They are still more valuable, perhaps, for that large class of students who cannot avail themselves of a teacher's guiding hand." Nature-" By making a judicious use of the examples, the student will find himself materially helped, especially if he is studying the subject without the aid of a teacher. We may also recommend this Key to teachers, who will find much of their time saved by having it in their possession." HIGHER ALGEBRA A SEQUEL TO ELEMENTARY ALGEBRA FOR SCHOOLS 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 FIFTH EDItion, revised and enlarged London MACMILLAN AND CO., LIMITED NEW YORK: THE MACMILLAN COMPANY IN this edition the text and examples are substantially the same as in previous editions, but a few articles have been recast, and all the examples have been verified again. We have also added a collection of three hundred Miscellaneous Examples which will be found useful for advanced students. These examples have been selected mainly but not exclusively from Scholarship or Senate House papers; much care has been taken to illustrate every part of the subject, and to fairly represent the principal University and Civil Service Examinations. OPINIONS OF THE PRESS Athenæum-"The Elementary Algebra by the same authors, which has already reached a third edition, is a work of such exceptional merit that those acquainted with it will form high expectations of the sequel to it now issued. Nor will they be disappointed. Of the authors' Higher Algebra as of their Elementary Algebra, we unhesitatingly assert that it is by far the best work of its kind with which we are acquainted. It supplies a want very much felt by teachers." Academy-" Is as admirably adapted for College students as its predecessor was for schools. It is a well-arranged and well-reasoned-out treatise, and contains much that we have not met with before in similar works. For instance, we note as specially good the articles on Convergency and Divergency of Series, on the treatment of Series generally, and the treatment of Continued Fractions. The book is almost indispensable, and will be found to improve upon acquaintance." Saturday Review-"They have presented such difficult parts of the subject as Convergency and Divergency of Series, Series generally, and Probability with great clearness and fulness of detail. No student preparing for the University should omit to get this work in addition to any other he may have, for he need not fear to find here a mere repetition of the old story. We have found much matter of interest and many valuable hints. We would specially note the examples, of which there ... are enough, and more than enough, to try any student's powers.' School Guardian-"We have no hesitation in saying that, in our opinion, it is one of the best books that have been published on the subject. . . . The last chapter supplies a most excellent introduction to the Theory of Equations. We would also specially mention the chapter on Determinants and their application, forming a useful preparation for the reading of some separate work on the subject. The authors have certainly added to their already high reputation as writers of mathematical text-books by the work now under notice, which is remarkable for clearness, accuracy, and thoroughness. Although we have referred to it on many points, in no single instance have we found it wanting." ... "It is a splendid sequel to your Elementary Algebra, and I am very pleased to see you have introduced the essential parts of the Theory of Equations in Chap. XXV., which contains all that is required of the subject for ordinary practical purposes." A. G. GREENHILL, M.A., Professor of Mathematics to the Senior Class of Artillery Officers, R.A. Institution, Woolwich. 94 HIGHER ALGEBRA. An independent variable is a quantity which may have any value we choose to assign to it, and the corresponding dependent variable has its value determined as soon as the value of the independent variable is known. *123. An expression of the form -2 P1"+P1x"1 +P ̧2 + ... + P+P where n is a positive integer, and the coefficients Po, P1, PP. do not involve a, is called a rational and integral algebraical function of x. In the present chapter we shall confine our attention to functions of this kind. *124. A function is said to be linear when it contains no higher power of the variable than the first; thus ax + b is a linear function of x. A function is said to be quadratic when it contains no higher power of the variable than the second; thus ax2 + bx + c is a quadratic function of x. Functions of the third, fourth,... degrees are those in which the highest power of the variable is respectively the third, fourth,.... Thus in the last article the expression is a function of x of the nth degree. *125. The symbol f(x, y) is used to denote a function of two variables x and y; thus ax+by+c, and ax2+ bxy + cy3 + dx + ey +ƒ are respectively linear and quadratic functions of x, y. The equations f(x) = 0, ƒ (x, y) = 0 are said to be linear, quadratic,... according as the functions f(x), f(x, y) are linear, quadratic,.... *126. We have proved in Art. 120 that the expression ax2+ bx + c admits of being put in the form a (x-a) (x - ẞ), where a and ẞ are the roots of the equation ax2 + bx + c = 0. Thus a quadratic expression ax + bx + c is capable of being resolved into two rational factors of the first degree, whenever the equation ax2 + bx + c = 0 has rational roots; that is, when b2-4ac is a perfect square. may *127. To find the condition that a quadratic function of x, y be resolved into two linear factors. Denote the function by f(x, y) where ƒ (x, y) = ax3 +2hxy + by3 + 2gx+2ƒy+c. 248 HIGHER ALGEBRA. *305. The series whose general term is if p>1, and divergent if p= 1, or p< 1. By the preceding article the series will be convergent or divergent for the same values of p as the series whose general fore the given series will be convergent or divergent for the same values of p as the series whose general term is required result follows. [Art. 290.] 1 Hence the *306. The series whose general term is u is convergent or di vergent according as Lim [{n(-1)-1} log n]>1, or < 1. Let us compare the given series with the series whose general When p>1 the auxiliary series is convergent, and in this case the given series is convergent by Art. 299, if |