ERRATA. Page 24, for 1. 13, read By (7) dx du cot.u cosec.u. du xa)2, read (1 − x2)3. 29, 1. 4, for etan.x, read etan.—'x. = -dx cot.u cosec.u 100, l. 12, for (1 — y22)a, read (1 — yɔj3⁄4. 1. 20, read.. (2. 54. Ex. 4)u 103, 1. 7, for l, read l. x3sin.-1x √1−x2(x2 + 2) = 3 9 147, 1. 10 from bottom, for 3xyf." + y3f!',,, read 3xy'ƒ.',, + y3ƒ• • 184, 1. 9 from bottom, for read 1 202, 1. 4, for 2a3, read 2a3x. 206, 1. 2, for b6a, read 6x2. 225, 1. 3 from bottom. The chapter here referred to is removed to the second volume. 239, 1. 17. The N in this and the two following lines is to be multiplied by abc. 257, Praxis 2. This curve has four conjugate points, which are the only points that belong to the curve. The equation may be considered as a particular case of (x2- a2)2 + (y2 — b2)2=c1. The equation of two ellipses whose axes are at right angles to each other is (a2x2)(b2- x2). THE ELEMENTS OF THE FLUXIONAL CALCULUS.. PART FIRST. Explanation of Terms. 1. ALL quantity, whether represented by lines and figures, or expressed algebraically, may be considered as generated by motion lines, by the motion of a point; areas, by a line moving parallel to itself; solids, by the motion of an area along a fixed line, which is the axis of the solid. When quantity is expressed algebraically, the first letters of the alphabet are usually taken to denote the constant, and the last the variable part. In curves, the ordinate and abscissa are variable lines; and the diameters, axes, parameters, &c. are, in general, considered as constant quantities. 2. A function of a variable quantity is any expression of calculation whatever into which the variable, mixed or not with constant quantities, enters. sin. Thus x", a, log. x, ax + b, m+nx+cx2, 1+ax+bx2 1+ax+ßx22 x, sin. cos., &c. &c. are all functions of r. Symbols to represent these are Fx, fx, qx, 4x, &c.; thus, 1+ ax + bx2 if in any particular case, Fa represents 1+ ax + ßx2 B and fx bx + cx2)" + bx + cr2 may be put under the form of a function of a + bx + cx2, for it equals (a + bx + cx2)" + (a + bx + cx2) a. The ordinate of a curve is a function of its abscissa. The space described by a body projected downwards is a function of the time of descent, the velocity of projection entering the function as a constant. And, generally, if it can be shown of two quantities m and n, that m varies when n varies, and that m is constant when n is constant, then we know that m = on, where the form of 4 is to be determined from the conditions of the question. If in the symbol far we make x = 0, it becomes f., which, therefore, represents a constant quantity, or rather a function in which a does not enter. Thus, if fx = ax + b, f. = b; When the function is enclosed in a bracket, and no other reason appears, it expresses the function when a particular value is assigned to the variable. Analytical functions are either algebraick or transcendental. The former are subdivided into rational and irrational: the latter, into exponential, logarithmick, and circular. Instances of transcendental functions are a", log. x, sin. x, ab. cos. x, &c. x3-axy + y3 is a function of two variables, and general symbols are F(x, y), f(x, y), &c. If y2 xy-a2 = 0, y is said to be an implicit function of a; but when this equation is solved with respect to y, or when we put it under the form x2 y = + + a2, 2 4 Symbols for y is then said to be an explicit function of x. each of these are F(x, y) = 0, and y = fx. 3. F(fr) represents a function of a function of r; thus, if fx = (a+bx + cx2)”, F(ƒx) may represent (a + bx+cx2)mn, or e(a+b+c)", &c. &c. If F =ƒ, ƒ(fx) is denoted by f2x, ƒ(ƒ2x) by ƒ3x, &c.; and, according to this notation, sin. x does not represent square of the sine of x, but the sine of the sine of x, i. e. the sin. x being formed into an arc, its sine is the line represented by sin.*. Also log.x = log. of log. x. We must be careful to distinguish fer from the square of fr or fr2; thus, let fra3, then fr, in this notation, = (x3)3 = x9, but fr2 = x6. = = 4. The variable which is under the sign of the function in y = fx is called the principal or independent variable; the other the dependent variable, as it depends upon the value we assign to x. 5. If the independent variable be increased or diminished, and the function can be expanded, we can compare the increments of the variables. As an example, take y=x3, and when a becomes x+h, let y become y, then y = (x + h)3 = x3 + 3x2h+3xh2 + h3, therefore Yy = 3.x h + 3xh2 + h3, and dividing by h, Y-y h inc. x. = 3x2 + 3xh + h2, which is the value of inc. y: The first term of this result is independent of h; if then and we have = 3x2, the mean we suppose h = 0, y = y, 0 0 6. In the present work we consider all such functions as flowing quantities or fluents, and the two problems which include the whole of the fluxional calculus are, (1). Given the fluents, to find their rates of increase; and, (2). Given the rates of increase, to find the fluents. The first is called the direct, the second the inverse method of fluxions. *We shall not make use of this notation in "circular" functions without giving the reader notice. |