5. The correlative term to derived is fundamental. Thus, when the units of area, volume, velocity, and acceleration are defined as above, the units of length and time are called the fundamental units.

Dimensions.

6. Let us now examine the laws according to which derived units vary when the fundamental units are changed.

Let V denote a concrete velocity such that a concrete length L is described in a concrete time T ; and let v, l, t denote respectively the unit velocity, the unit length, and the unit time.

The numerical value of V is to be equal to the numerical value of L divided by the numerical value of V L Τ

T. But these numerical values are

(1)

ย / T

This equation shows that, when the units are changed (a change which does not affect V, L, and T), v must vary directly as and inversely as t; that is to say, the unit of velocity varies directly as the unit of length, and inversely as the unit of time.

V

Equation (1) also shows that the numerical value of a

ข

given velocity varies inversely as the unit of length, and directly as the unit of time.

7. Again, let A denote a concrete acceleration such that the velocity V is gained in the time T', and let a denote the unit of acceleration. Then, since the

numerical value of the acceleration A is the numerical value of the velocity V divided by the numerical value of the time T, we have

This equation shows that when the units a, l, t are changed (a change which will not affect A, L, T or T'), a must vary directly as 7, and inversely in the duplicate

A

ratio of t; and the numerical value will vary inversely

a

as, and directly in the duplicate ratio of t. In other words, the unit of acceleration varies directly as the unit of length, and inversely as the square of the unit of time; and the numerical value of a given acceleration varies inversely as the unit of length, and directly as the square of the unit of time.

It will be observed that these have been deduced as direct consequences from the fact that [the numerical value of] an acceleration is equal to [the numerical value of] a length, divided by [the numerical value of] a time, and then again by [the numerical value of] a time.

The relations here pointed out are usually expressed by saying that the dimensions of acceleration* are

length

(time)2'

that the dimensions of the unit of acceleration* are

or

* Professor James Thomson ('Brit. Assoc. Report,' 1878, p. 452) objects to these expressions, and proposes to substitute the following:-

unit of length

(unit of time)2*

8. We have treated these two cases very fully, by way of laying a firm foundation for much that is to follow. We shall hereafter use an abridged form of reasoning, such as the following:

Such equations as these may be called dimensional equations. Their full interpretation is obvious from what precedes. In all such equations, constant numerical factors can be discarded, as not affecting dimensions.

9. As an example of the application of equation (2) we shall compare the unit acceleration based on the foot and second with the unit acceleration based on the yard and

minute.

Let / denote a foot, L a yard, t a second, T a minute, Ta minute. Then a will denote the unit acceleration based on the foot and second, and A will denote the unit acceleration based on the yard and minute. Equation (2) becomes

that is to say, an acceleration in which a yard per minute

Change-ratio of unit of acceleration=

change-ratio of unit of length", (change-ratio of unit of time)2

This is very clear and satisfactory as a full statement of the meaning intended; but it is necessary to tolerate some abridgment of it for practical working.

I

of velocity is gained per minute, is of an acceleration

1200

in which a foot per second is gained per second.

Meaning of "per."

10. The word per, which we have several times employed in the present chapter, denotes division of the quantity named before it by the quantity named after it. Thus, to compute velocity in feet per second, we must divide a number of feet by a number of seconds.*

If velocity is continuously varying, let x be the number of feet described since a given epoch, and t the number of seconds elapsed, then is what is meant by the

dx

dt

number of feet per second. The word should never be employed in the specification of quantities, except when the quantity named before it varies directly as the quantity named after it, at least for small variations—as, in the above instance, the distance described is ultimately proportional to the time of describing it.

Extended Sense of the terms "Multiplication" and
"Division."

11. In ordinary multiplication the multiplier is always a mere numerical quantity, and the product is of the same nature as the multiplicand. Hence in ordinary division either the divisor is a mere numerical quantity and the quotient a quantity of the same nature as the dividend;

* It is not correct to speak of interest at the rate of Five Pounds per cent. It should be simply Five per cent. A rate of five pounds in every hundred pounds is not different from a rate of five shillings in every hundred shillings.

or else the divisor is of the same nature as the dividend, and the quotient a mere numerical quantity.

But in discussing problems relating to units, it is convenient to extend the meanings of the terms "multiplication" and "division." A distance divided by a time will denote a velocity-the velocity with which the given distance would be described in the given time. The distance can be expressed as a unit distance multiplied by a numerical quantity, and varies jointly as these two factors; the time can be expressed as a unit time multiplied by a numerical quantity, and is jointly proportional to these two factors. Also, the velocity remains unchanged when the time and distance are both changed in the same ratio.

all denote the same velocity, and are therefore to be regarded as equal. In passing from the first to the second, we have changed the units in the inverse ratio. to their numerical multipliers, and have thus left both the distance and the time unchanged. In passing from the second to the third, we have divided the two numerical factors by a common measure, and have thus changed the distance and the time in the same ratio. A change in either factor of the numerator will be compensated by a proportional change in either factor of the denominator.