branches of the subject, and for a corresponding variety in the mode of demonstration. As my sole object has been to produce a work of utility, I have not hesitated to avail myself of existing elementary works; of those to which I am chiefly indebted a list is subjoined. My thanks are due to many friends who have taken an interest in the book, and who have favoured me with valuable suggestions and corrections which have relieved it from various imperfections it would otherwise have contained. ST JOHN'S COLLEGE, I. TODHUNTER. Cournot, Traité élémentaire de la Théorie des Fonctions et du Calcul Infinitésimal, 2 vols. 8vo. Paris, 1841. De Morgan, Differential and Integral Calculus. London, 1842. Duhamel, Cours d'Analyse de l'Ecole Polytechnique, 2nd edit., 2 vols. 8vo. Paris, 1847. Moigno, Leçons de Calcul Différentiel et de Calcul Intégral, 2 vols. 8vo. Paris, 1840-44. Navier, Résumé des Leçons d'Analyse données à l'Ecole Polytechnique, 2 vols. 8vo. Paris, 1840. Schlömilch, Handbuch der Differenzial- und Integralrechnung. Griefswald. 1847. XIII. Maxima and minima values of functions of one variable Students reading this work for the first time may omit the following 23, 83, 98-113, 141-143, 151, 156, 163-166, 185-189, 200, DIFFERENTIAL CALCULUS. CHAPTER I. DEFINITIONS. LIMIT. INFINITE. 1. SUPPOSE two quantities which are susceptible of change so connected that if we alter one of them there is a consequent alteration in the other, this second quantity is called a function of the first. Thus if x be a symbol to which we can assign different numerical values, such expressions as 2, 32, log x, and sin x, are all functions of x. If a function of x is supposed equal to another quantity, as for example sin xy, then both quantities are called variables, one of them being the independent variable and the other the dependent variable. An independent variable is a quantity to which we may suppose any value arbitrarily assigned; a dependent variable is a quantity the value of which is determined as soon as that of some independent variable is known. Frequently when we are considering two or more variables it is in our power to fix upon whichever we please as the independent variable, but having once made our choice we must admit no change in this respect throughout our operations; at least such a change would require certain precautions and transformations. 2. We generally denote functions by such symbols as F(x), f(x), (x), (x), and the like, the variable being denoted by x. Such an equation as y(x) implies that the dependent variable y is so connected with the independent variable, that the value of y becomes known as soon as that of x is given, and that if any change be made in the numerical value assigned to x, the consequent change in y can be found. T. D. C. B 3. The student has probably already had occasion to consider the meaning of the terms "variable quantity" and "function" which we have here introduced. In treatises on the conic sections, for example, the equation y = 2 Jax occurs, y=2√ax where x is a general symbol to which different numerical values may be assigned, and a is a symbol to which we suppose some invariable numerical value assigned, and which is therefore called a "constant." For every value given to x we can deduce the corresponding numerical value of y. In the equation y = 2 Jax, since the value of y depends upon that of a as well as that of x, we may say that У is a function of a and x. Hence such symbols may be used as F(a, x) to denote a function of both a and x, and such an equation as y = (x, z, t) indicates that y is a function of the three quantities denoted by the symbols x, z, and t. 4. In the equation y = 2 Jax, if we know that a is to be a constant quantity throughout any investigation on which we may be engaged, we shall frequently not require to be reminded of this constant, and shall, continue to speak of y b a as a function of x. So the equation y = √(a2 — x2) may be represented by y=(x), where we express only that symbol x which throughout our investigations will be considered variable. 5. If the equation connecting the variables x and y be such that y alone occurs on one side and on the other side some expression involving x and not y, we say that y is an explicit function of x. When an equation connecting x and y is not of this form, we say that y is an implicit function of x. Thus if y = ax + bx + c, we have y an explicit function of x. If ay2 - 2bxy + cx2+g=0, y is an implicit function of x. The words implicit function assume that y really is a function of x in the sense in which we have used the word function. This assumption may be seen to be true in the example given, for we can by the solution of a quadratic equation exhibit y as a function of x; or rather we can infer that y must be one of two explicit functions of x, namely bx+√√ { (b2 — ac) x2 — ag} _bx-√/{(b2 — ac) x2 — ag} either a or a shall return to this point hereafter, in Art. 58. We |