fixed end, and the velocity of this moveable end is therefore w. Thus we have w=x,

This rate of change of a fluent quantity is called its fluxion, or sometimes, more shortly, its flux. It appears from the above considerations that a flux is always to be conceived as a velocity; because a quantity must be continuous to be fluent, must therefore be specified either by a line or an angle (which may be placed at the centre of a standard circle and measured by its arc) and rate of change of a length measured on a straight line or circle means velocity of one end of it (if the other be still) or difference of velocity of the two ends.

The flux of any quantity is denoted by putting a dot over the letter which represents it.

If a variable angle aop be placed at the centre of a circle of radius unity, and the leg oa be kept still; the velocity of p will be the flux of the circular measure of the angle (since ap: oa circular measure, and oa = 1). This is called the angular velocity of the line op. When the angular velocity is uniform, it is the circular measure of the angle described in one second.

Ө

When one end of a vector is kept still, the flux of the vector is the velocity of the other end. Thus if p represent the vector from the fixed point

o to the moving point p, p is the velocity of p. But when both ends move, the

flux of the vector is the difference of their velocities. Thus if

The rate of change of the vector ab is the velocity of b compounded with the reversed velocity of a.

α

B

α

DERIVED FUNCTIONS.

When two quantities are so related, that for every value of one there is a value or values of the other, so that one cannot change without the other changing, each is said to be a function of the other. Thus every fluent quantity is a function of t, the number of seconds since the beginning of the time considered. For example, in parabolic motion, the position-vector p=a+tẞ+ty is a function of the time t. Here the function is said to have an analytical expression of a certain form, which gives a rule for calculating p when t is known. A function may or may not have such an expression.

A varying quantity being a certain function of the time, its flux is the derived function of the time. Thus if = a + tẞ+ty, we know that p = B+ 2ty. Then ẞ+2ty is the derived function of a+t3+ty. When a function is rational and integral, we know that the derived function is got by multiplying each term by the index of t, and then diminishing that index by 1. We proceed to find similar rules in certain other cases.

The flux of a sum or difference of two or more quantities is the sum or difference of the fluxes of the quantities. This is merely the rule for composition of velocities.

Flux of a product of two quantities. Let p, q be the quantities, and let p1, q, and p, q, be their values at the times t, and t, respectively. Then we have to form the quotient p11 - P2l2 t1t, cast out common factors from numerator and denominator, and then omit the suffixes. Now

but when we cast out common factors and omit the suffixes from the latter expression, it becomes på+pq. Thus the flux of a product is got by multiplying each factor by the flux of the other, and adding the results.

FLUX OF PRODUCT AND QUOTIENT.

65

This is equally true when both the factors are scalar quantities, and when one is a scalar and the other a vector. We cannot at present suppose both factors to be vector quantities, because we have as yet given no meaning to such a product.

When both factors are scalar, this result may be written in a different form. Let u=pq, then u=pq+ pq. Divide by u, then

+

v = pqr+pår+pqr;

and it is clear that this theorem may be extended to any number of factors.

:

Flux of a quotient of two quantities. Let p q be the quotient; then we have

and the latter expression, when we cast out common factors and omit the suffixes, becomes pq - på: q2. If we write up q, then u = pq-pq: q', or dividing by u, that is multiplying by q and dividing by p, we find

from which a formula for the quotient of one product by another may easily be found.

We might of course use any other letter instead of t to represent the time; and when an analytical expression is

m

given us, involving two or more letters, we may find its derived function in regard to any one of them. Thus of the quantity u = x2+5y2+3xy, if x represents the time, the derived function is 2x + 3y; but if y represents the time, the derived function is 10y+3x. If we suppose x and y to be horizontal and vertical components of a vector op=xi+yj, then for every point p in the plane there will be a value of x, a value of y, and consequently a value of u, = x2+5y2+3xy. If we make the point p move horizontally with velocity 1 centimeter per second, then x will represent the time, and y will not alter; so that u will be 2x+3y. This is called the flux of u with regard to x, or the x-flux of u; and it is denoted by u. Similarly if we make p move vertically with the unit velocity, a will be constant, and y will represent the time, so that u will be 3x+10y; this is called the flux of u with regard to y, or the y-flux of u, and is denoted by a,u. The characteristic a may be supposed to stand for derived function.

We may now prove a very general rule for finding fluxes, namely one which enables us to find the flux of a function of functions. Let x and y be two variable quantities, and let it be required to find the flux of u which is a function of x and y; this is denoted thus: u =ƒ (x, y). The method is the same as that used for a product. We find

=

t1-to

f(x, y ) − f ( 3 ) f (2, 3) - f(x, y,)
2'
+

t1-t

t1-t2

and when we strike out common factors and omit the suffixes in this last expression, it becomes f+ÿf;

where ƒ has been shortly written instead of ƒ (x, y). Or, substituting u for f, we have the formula

If a straight line ov be drawn through a fixed point o, to represent in magnitude and direction at every instant the velocity of a moving point p, the point v will describe some curve in a certain manner. This curve, so described, is called the hodograph of the motion of p. (ódòv ypáþei, it describes the way.)

b

P

Thus in the parabolic motion p = a+tẞ+ty, we have ov=p=B+2ty. Hence we see that the point v moves uniformly in a straight line. The hodograph of the parabolic motion, then, is a straight line described uniformly. Let ab be the initial velocity; draw through b a line parallel to the axis of the parabola. Then to find the velocity at any point p, we have only to draw av parallel to the tangent at p; the line av represents the

a

velocity in magnitude and direction. The straight line bv, described with uniform velocity 2y, is the hodograph. In the elliptic harmonic motion

we have

= a cos (nt + e) + ß sin (nt + e)

p =

ov = p = na cos (nt + e + 1⁄2π)+nẞ sin (nt + e + 1⁄2π). Thus the point v moves harmonically in an ellipse similar and similarly situated to the original path, of n times its linear dimensions, being one quarter phase in advance. As a particular case, the hodograph of uniform circular mo