From (2) and (3) we have, by subtraction, (4) Dy = F(x + Dx) — .F(x); whence we have the ratio Dy _11x+Dx)—Fx) Dx Dx This ratio will generally change in value as Dx and Dy diminish, till, as they both vanish, which they must do simultaneously, it assumes the form Q• Takingthis form, it ceases to have a determinate actual value, and it is necessary to resort to the method of limits, to ascertain the value to which it was approaching, as Dx and Dy approached zero. Let, then, dx and dy be any quantities whose ratio is equal to the limiting ratio of the increments Dx, Dy, so that dy Dx = limit Dy dx as Dx and Dy approach zero. Then dx and dy are the differentials of x and y. It may be observed that where x and y are connected as above, they cannot vary independently of one another. In the case assumed, x has been taken as what is called the independent variable, the question being, how does y vary when x varies. If y were made the inde pendent variable, it would be necessary to solve the equation y = F(x), if possible, so as to express x in terms of y. The result would be an equation x = 9(y). This being D obtained, we should find x . = limit as before. It will be seen that on this view differentials are defined merely by their ratio to one another. Their actual magnitude is perfectly arbitrary. This, however, does not render an equation involving differen tials indeterminate, since their relative magnitude is definite, and since, front the nature of the definition, a differential cannot appear on one side of an equation without another connected with it appearino. on the other.
The idea of a differential being once comprehended, the reader will be able to under stand, in a general way, the main divisions of the C., which we shall now briefly delin eate. So much is clear from what has been stated, that there must be two main divisions —one by which, the primary quantities being known, we may determine their differen tials; and another by which, knowing the differentials, we may detect the primary quantities. These divisions constitute the differential C. and integral C. respectively.
1. Till: DIFFERENTIAL CALCULUS.—Recurring to the formula already given we know dyDy + Dx) — 11x) dx =limit = limitDx Dx • It is clear that, in the general case, F(x + D — Ilx) at the limit will still be some function of x. Calling it F' (x), we have generally = F' F' (x) is called the first differential coefficient of y or .1 z). Being a function of x, it may be again differenti
ated. The result is written — = F"(x), de F"(x) being the second differential coefficient of y or Ilx); and again F"(x) may be a function of x, and so capable of differentiation. Now it is the object of the differential C. to show how to obtain the various differentials of those few simple functions of quantity which are recognized in analysis, whether they are presented singly or in any form of combination. Such functions are the sum, difference, product, and quotient of variables, and their powers and roots; exponentials, logarithms; and direct and inverse circular functions. The C. so far is complete as we can differentiate any of those func tions or any combination of them—whether the functions be explicit or implicit; and with equal ease we may differentiate them a second or any number of times. This C.
is capable of many interesting applications as to problems of maxima and minima, the tracine., of curves, etc., which cannot here be particularly noticed.
2. 'Tim INTEGRAL CALCULUS deals with the inverse of the former problem. The former was. Given FM, to find If''(x), F"(x), and so on. The present is in the simplest case—viz., that of an explicit function. Given di/ — = to find The methods of the integral C., instead of being general, are little better than artifices suited to particular cases; no popular view can be given of these. In many cases, integration is quite impossible. The explanation of integration by parts, by approximation, definite Integrals, and singular solutions, is far beyond the scope of the present work. The reader is referred to any of the numerous textbooks on the subject. The integral C. has appli cations in almost every branch of mathematical and physical science. It is specially of use in determining the lengths of curved lines, the areas of curved surfaces, and the solid contents of regular solids of whatever form. The whole of the lunar and planetary theories may be described as an application of the integral C., especially of that branch of it which deals with the integration of differential equations It is applied, too, in hydrostatics and hydrodynamics, and in the sciences of light, sound, and heat. In short, it is an instrument without which most of the leading triumphs in physical science could never have been achieved.