and is supposed to pass the origin at the instant when P passes, at which point x=0J=1. relatively to that of Q, or vice versa, is as soon as the ratio of μ to λ is given. construction the terms modulus, base, logarithm are defined as follows:

The speed of P obviously known By means of this exponential, and

(i). μ/λ, a given value of which, say m, determines a system of corresponding distances x', x" andy', y",...

is called the modulus of the system.

(ii).

The modulus having been assigned, the value of x, corresponding to y=0J1, is determined as a fixed magnitude, and is called the base of the system. Let it be denoted by b.

(iii). x, x', x'', . . . are called the exponentials of y, y′, y", respectively, with reference either to the modulus m or the base b, and the relation between x and y is written

x'',

x= expm 1', or x=b***

(iv). _y, y', y'', ... are called the logarithms of x, x′, respectively, with reference to the modulus m, or the base b, and the symbolic statement of this definition is either

y=log, x, or y log x.*

The convention that P shall be at J when Q is at O introduces the convenient relations

logm I

=

o, and expm 0 = I, or b°

and because (by definition) y I when x =

*

... logm bì, and expm I=

7

b,

I,

= b, or b1 = b.

* blog x is the German notation. English and American usage has hitherto favored writing the base as a subscript to log thus, log; but Mr. Cathcart in his translation of Harnack's Differential u. Integral Rechnung has retained the German form, which is here adopted as preferable.

**See Appendix, page 139.

Exponentials and logarithms are said to be inverse to each other.

(v). The logarithms whose modulus is unity are called natural logarithms, and the corresponding base is called natural base, the special symbols for which are In and e respectively; thus

represent the logarithm and its inverse, the exponential, in the natural system.

24. Relations between Base and Modulus. Let the speed of P remain equal to Xx as in Art. 23 while the speed of Q is changed from μ to kμ. The modulus, p/λ=m, will then be changed to kμ/λ=km, and the distance of Q from the origin, corresponding to the distance x, will become ky. Hence

that is: To multiply the modulus of a logarithm by any real quantity has the effect of multiplying the logarithm itself by the same quantity.

Corresponding to this relation between logarithms in the systems whose moduli are m and km, the inverse, or exponential relation is

* These are also sometimes called Napierian logarithms, but it is well known that the numbers of Napier's original tables are not natural logarithms. The relation between them is expressed by the formula

Napierian log of x=107 ln (107 | x).

Napier's system was not defined with reference to a base or modulus. See the article on Napier by J. W. L. Glaisher in the Encyclopædia Britannica, ninth edition.

Let c be the base in the system whose modulus is km, then

since y

when x =b, and

ky=logkm x = log km

M = K

C= 1, when x = c,

1

с

c = expm = b1/k ̧*

Hence: If the modulus be changed from m to km, the corresponding base is changed from b to b1⁄4k.

Again, if in the equation log, x = m ln x, b be substituted for x, the value of m is obtained in the form

m = 1 / lnb.

Hence in terms of the base of a system the logarithm is

and if to x the value e be given, m is obtained in the form

m= logm e;

whence passing to the corresponding inverse relation:

expm m = = bm e;

that is: The exponential of any quantity with respect to itself as a modulus is equal to natural base.

25. The Law of Involution. By virtue of the fundamental principle of the last article-that to multiply the modulus multiplies the logarithm by the same amountwe have in general

*For an amplification of this proof see Appendix, page 139.

or in other terms, since changing m into m/h, or m/k, changes into b, or b1⁄2,

or both, may be replaced by their reciprocals in this formula. and the law, more completely stated, is

(b×h)×,k = (b×,k) ׂh.

Evolution. If k be an integer, the process indicated by bik is called evolution; and, when k2, it is usually expressed by the notation √ b.

26.. The Law of Metathesis. Let z= = bh; then by the law of involution

which expresses the law of interchange of exponents with coefficients in logarithms,-the law of metathesis.

27. The Law of Indices.* The law of indices, or of addition of exponents, follows very simply from the definition of the exponential, thus: In the construction of Art. 23, and by virtue of Definition (iii) of that Article

*For an alternative proof see Appendix, pages 139-141.

V

But the magnitude 1+ P'P'' OP' represents the distance that P would traverse, starting at unit's distance to the left of P', on the supposition that its speed at P' is λx'x'=λ, and the corresponding distance passed over by Q is y"—y'; hence, by definition of the exponential,"

and therefore

1 + P'P'' | OP'= expm (y'' — y′),

expm y'' expm y' expm (y''-y′),

which is the law of indices for a quotient. In particular, if y' o

or

expm o/expm y' expm (o-y')

- =

1 / expm y'= expm (— y′);

(Art. 23.)

and therefore, writing -y'y, the foregoing equation, expm y''/ expm y' expm (y''-y'), becomes

expm y'' expm y=expm (y''+y),

which is the law of indices for a direct product.

The com

plete statement of this law is therefore embodied in the formula

expm y'' expm y':

expm. (y''±y′),

by" Xy by! = by" ="y", "

or its equivalent

28. The Addition Theorem. Operating upon both sides of the equation last written with logm we have

logm (by′′ × b1′)=y''±y′;

that is, replacing b", by by x'', x' and y'', y' by logm x”, logmx',

logm (x'' x') = logm x'' logm x′,

which is the addition theorem for logarithms.