tion we find that b' is also a pure number. In this problem, therefore, both horizontal and vertical ordinates can be nothing but pure numbers.*

At the same time it must be noted that if k be a physical constant quantity, either a unit or any other quantity, then k represents a varying physical quantity of the same kind.

95. Power Gradient of Exponential Function.-If n be any fixed number, " is also a fixed constant number. We e may write b2 = bnb (ln)

Here is the vertical height of the curve, fig. 20, at any horizontal distance 7; while b-n) is the height of a point on the curve at the constant horizontal distance n to the left of 1.

Considering various l's and various pairs of points with ordinates l and (-n), we perceive that the nature of this curve gives a constant proportion, 7", between the heights of all pairs of points at the constant horizontal distance n apart. A succession of points at the equal horizontal spacing n have their heights advancing in geometrical series, the common ratio of which is b". Taking the l-gradient, or slope, of the curve at l, and remembering that b" is a constant factor, we find

This means that there is also the same constant proportion, b”, between the gradients as between the heights at all pairs of points horizontally n apart.

Dividing the height by the slope we obtain the subtangent; see § 38.

Jdb

Subtangent at 1=say T='/alb

Subtangent at (1 — n) = say Tu-n) = b(-n)

(-n)

dl

These two last expressions are equal, because the common factor b" cancels out in their ratio. Thus the two subtangents are equal in length; see fig. 20.

Now the length of the subtangent at l does not depend in any way upon the length n. For any one point 1, we may take various lengths n, thus getting various points (-n), at all of which the subtangent equals that at the one point 7. This means that at all points along the curve the subtangent has the same length.

* In Sir William Hamilton's Quaternions, (-1) raised to a continuously increasing power is used to indicate rotation or gradual change of direction.

the constant being the reciprocal of the constant length of subtangent. The subtangent being a pure number, its reciprocal is also a pure number.

96. Natural, Decimal, and other Logarithms.-When T=1 (unity), and the above constant is therefore also unity, the base b is designated by mathematicians by the letter e. It, e, is the base of the "natural" or "hyperbolic" or "Neperian" logarithms.

Each system of logarithms has a "base," which is a number and to which all other numbers are referred. Every number, whether whole or fractional, is represented as this "base," raised to a certain power. The power to which the base must be raised in order to produce each number is the logarithm of that number. Thus, if b' be a number, and if b be taken as the base of the system of logarithms, then 7 is the log of the number 6' to the base b.

In the "Common," or "Brigg's," or "Decimal" logarithms, the base is 10; every number being represented as 10 raised to one or other power.

The result of § 95 may be expressed thus:-In any system of logarithms the logarithm-gradient of the number bears a constant proportion to the number itself. The reciprocal of this constant proportion, or the subtangent in the graphic representation of fig. 20, is called the "modulus" of the system of logarithms.

The "natural" system of logs may be defined as that which gives the rate of increase of the number compared with its logarithm equal to the number itself; or which makes the logarithmgradient of the number equal to the number itself; or, symbolically,

the constant for this base e being unity. The base e is calculated to be 2.71828 -- The modulus of this system is 1.

97. Number Gradient of Logarithm.--If we call the number N, then in fig. 20 the vertical height of the curve is N and the horizontal ordinate is its logarithm. The curve is sometimes termed a logarithmic curve; sometimes an exponential curve. It takes different forms according to the logarithmic base b used in drawing it out. In fig. 21 these variations of form are shown, the five curves having the bases

b=2, e, 8, 10, and 12.

All these curves come to the same height N=1 at 7=0. Atl=1 the height of each is the base b. To the left of the vertical axis where I is negative, the height N is always less than unity, and decreases asymptotically to the horizontal axis towards zero height. Fig. 22 shows the extension of these same curves to high numbers with positive logarithms. The logarithm to the base e of the number N is written log, N. The logarithm to the base b of N is written log, N. Thus the decimal log of N may be unambiguously written log10 N.

The results of §§ 95 and 96 written in this notation, and taking the reciprocal or conjugate gradient from the vertical axis, instead of the l-gradient, are

the constant T being the modulus of the system of logarithms; and

dlog N dl 1
dN dN

N

if natural logs are used in which the base is e and the modulus unity.

98. Relation between Different Log "Systems."--Any base b may be looked on as the base e raised to a certain power, which

we will call

Therefore, if

also

1

T'

log, N=1=Tlog, N=log, N. log, e.

Thus there is a constant ratio T-log, e between the logarithms of numbers to the base b to their logarithms in the "natural" system.

Taking the N-gradients of these last logs and remembering that T is a constant,

d log, N
dN

which shows that the T of this paragraph means the same as the T of the last, § 97, or the subtangent of fig. 20.

!

It is useful to notice that a logloga, because taking logs on each side, we find log, x, log, a=log, a. log, x.

99. Base of Natural Logs.-Although the calculation of e, or of the modulus T of any logarithmic system, is very laborious, there is no other difficulty about it beyond its tediousness. Thus to find the modulus of, say, the decimal system, an extremely small root of 10 has to be extracted. Thus the square root may be extracted, say, 100 times over. This will give (1)100th power of 10, which is a number a very minute fraction over 1. Now the decimal logarithm of this number is (1)100, which is easy to calculate, and the logarithm of 1 is zero. The former is, therefore, the increment of the logarithm corresponding to the number increment from 1 to the (1)100th power of 10. If the ratio of this logarithm increment to the number increment be multiplied by the number, which is here 1, the product is the decimal log-modulus.

The laborious part of the operation is finding the (1)100th power of 10. Extracting only a higher root, the increments will be larger, and the calculation will not give so great accuracy. Thus, if the extraction of the square root be repeated only 10 times, the result of the calculation gives

T10-0434 294 116

whereas by more minute calculation its true value to 9 decimal places has been found to be

0-434 294 482.

The error is only 0.000 000 366.

100. Logarithmic Differentiation.-Taking again the function of § 92, and taking logs, we have

log X= log L + log M+ etc. - log P- etc.

and differentiating with respect to a, we have by $$ 84 and 98

which is the same result again as was found in § 92.

This method of differentiating products of functions is called "logarithmic differentiation."

101. Change of the Independent Variable.-The rule deduced in § 80 was written

[X'adx.

Now X'dx is the same as SX; so that the integral on the right hand may be written

xdX. The equation becomes then

| Xdx=Xx - - fxdx.

If adX is easier to find than

Xda, this forms a method of

facilitating the latter integration. Such a transformation is called a "change of the independent variable" or "substitution.” Fig. 23 is the graphic representation of this law. Taking the

integration between the points 1 and 2 of the curve xX, the transformation is written