PROJECTION OF VELOCITY.

61

one as the diameter of the circle to the major axis of its projection. Hence an interval can be taken such that the mean velocity of the projected motion for every included interval shall differ from ov less than by a proposed quantity; or ov is the instantaneous velocity of the projected motion.

For an example of the last proposition, we may consider the simple harmonic motion, which is an orthogonal projection of uniform circular motion on a line in its plane. The velocity of p is na where a is the radius of the circle, and it is in the direction tp. The horizontal component of this is the velocity of m. The horizontal component is

=- na sin aop

na sin (nt + €)

= na cos (nt + e + 1⁄2π).

Hence if

we find

s = a cos (nt + €),

s = na cos (nt +€ + 1⁄2π),

t

a

and the rule is the same as in circular motion. The velocity is evidently n.pm, by which representation the changes in its magnitude are rendered clear. The same result may be obtained by means of the tangent to the harmonic curve, p. 43.

The velocity in elliptic harmonic motion may be found either by composition of two simple harmonic motions, or directly by projection from the circle. We thus find that when

p = a cos (nt + €) + ẞ sin (nt + e), then p=na cos (nt + e + 1⁄2π)+ nẞ sin (nt + € + 1⁄2π),

or the rule is the same as in the last case or in uniform circular motion. The result may also be stated thus: the velocity at the point p is n times of the semiconjugate diameter.

t

In the parabolic motion p = a + tß+ty we may now see that p = B+ 2ty.

the rule being to multiply each term by the index of t and then reduce this index by unity. Thus we can always find the velocity when the position-vector is a rational integral function of t.

FLUXIONS.

A quantity which changes continuously in value is called a fluent. It may be a numerical ratio, or scalar quantity (capable of measurement on a scale); or it may be a directed quantity or vector; or it may be something still more complex which we have yet to study. In the first case the quantity, being necessarily continuous because it changes continuously, can only be adequately specified by a length drawn to scale, or by an angle; and we may always suppose an angle to be specified by the length of an arc on a standard circle. Let one end of the length which measures the quantity be kept fixed, then as the quantity changes the other end must move. The velocity of that end is the rate of change of the quantity. Thus we may say that water is poured into a reservoir at the rate of x gallons per minute. Let the contents of the reservoir be represented on a straight line, so that every centimeter stands for a gallon; and let the change in these contents be indicated by moving one end of the line. Then this end will move at the rate of x centimeters per minute. If w is the number of gallons in the reservoir, it is also the distance of the moveable end of the line from the

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.

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

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 p = a + tẞ+ty, we know that p = 8+2ty. Then B+2ty is the derived function of a +t+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, q1 and P2, 9, be their values at the times t, and t, respectively. Then we have to form the quotient p11 — P2l2 : t1 — t2, 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 pq+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 upȧ+ pq. Divide by u, then

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-pq q. If we write up q, then i = 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