a being a positive quantity, it is plain that y will be imaginary whenever x is of the form

A and μ being integers. Now between any two values of x, however little they may differ from each other, we may intercalate an infinite number of fractions of the above form. Thus we see that it is impossible to join by a continuous curve two points of the locus of the equation, corresponding to two systems of real values of x and y, however near they may be to each other. These anomalous functions are inapplicable to questions of natural philosophy, and have attracted but little attention even in pure analysis. In this treatise we shall direct our attention entirely to continuous functions.

8. Functions are distinguished also by the names of algebraical and transcendental. If y be connected with x by an equation involving only the ordinary operations of addition and subtraction, multiplication and division, evolution and involution of assigned degrees, y is said to be an algebraical function of x. If the equation of connection does not satisfy this condition, involving for instance exponential, logarithmic, or circular functions of x and y, then y is said to be a transcendental function of x. Thus, for examples, in the equations

y is an algebraical function of x; and a transcendental one

be the equation to a curve, y will for a certain value of x experience a dissolution of continuity by becoming impossible, although the curve is itself continuous at the point. Thus if

y2 = 4mx,

y will continuously vary from one value to another as ≈ decreases from any assigned positive value down to zero, but, the moment x becomes negative, y becomes impossible. The two branches corresponding to the double sign, each of which terminates abruptly at the origin, join together at this point and thus form a continuous curve.

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CHAPTER II.

PRINCIPLES OF DIFFERENTIATION.

SECTION I. GENERAL FUNCTIONS.

Definition of a Differential Coefficient.

10. LET y be a certain function of x, and let y' be the value assumed by y when x becomes x'. Then, as x' keeps continuously approaching to the value of x, the fraction

will continuously tend towards a certain value from which it will ultimately differ by a quantity less than any assignable magnitude, or, in other words, to which it will be ultimately equal. The indefinitely small values of the differences x - x, y' y, are usually denoted by the symbols Sx, Sy, and the ultimate value of the fraction

In this expression dx and dy are any quantities whatever, either finite or infinitesimal, which are in the ratio of the

dy

ultimate values of 8x and dy. The fraction is called the

dx

differential coefficient of y with regard to x, the quantities dx and dy being called the differentials of x and y. The object of the Differential Calculus is to investigate the pro

perties of differentials and differential coefficients, and to develop the general principles of their application to the theory of coordinate geometry and other branches of pure mathematics, and to the estimation of the phenomena of nature.

approaches indefinitely near to x, the left-hand

When

that is, the differential coefficient of x with respect to x is 3x2, and its differential is 3x2dx.

Differentiation of a Constant.

11. If y = c, where c denotes any constant quantity, that is, any quantity which does not experience variation in consequence of a variation in the value of x, then

or the differential coefficient of a constant quantity is always

zero.

Differentiation of the Sum of a Function and a Constant.

12. If u = y + c, where y represents any function of x, and c denotes a constant quantity, then

du dy

=

dx dx

Let x', y', u', be simultaneous values of x, y, u; then

or the differential coefficient of the sum of a function and a constant is the same as that of the function alone.

Differentiation of the Product of a Function and a Constant. 13. If u = cy, where c represents a constant quantity and y a function of x; then

Let x', y', u', be simultaneous values of x, y, u; then

or the differential coefficient of the product of a function and a constant is equal to the product of the constant and the dif ferential coefficient of the function.