method of fig. 17, drawing a curve with ordinates X and E, and considering the increment of the rectangular area XE. This increment is evidently

X.SE+E.SX

and dividing this by dx the above result is obtained. Written as an integration this result is

This latter form is the most fundamental and useful of the "formulas of transformation," and is usually referred to as "Integration by Parts." By its help most of the "formulas of reduction" are obtained.

88. Product of any Number of Functions.-This result may at once be extended to the product of any number of different functions X, E, X, etc. etc. of x. The x-gradient of this product is the sum of a number of terms, each of which is the product of all but one of the factors multiplied by the x-gradient of that one factor omitted. The result may be written in more symmetrical form if each term be divided by the product of all the factors. Thus, call the whole product X, or let

then

X=E, etc., etc.

+ + etc. + etc.
XEX
我

where X', E', ', etc., are the x-gradients of X, E, A, etc.

89. Reciprocal of Product of Two Functions.-The x-gradient of the reciprocal of the product of two functions E and of x is found with equal ease. Call this reciprocal X and its x-gradient

90. Reciprocal of Product of any Number of Functions.-The extension of this law to the reciprocal of the product of any number of functions is made by simply repeating the process of the last paragraph.

91. Ratio of Two Functions.-To find the x-gradient of the quotient of two functions of x, we combine the laws of § 87 and § 86. Thus, if

It will be a useful exercise for the student to deduce this result directly by the method of fig. 18 and § 81, making the co-ordinates to the curve E and instead of X and x, and without using the law

92. Ratio of Product of any Number of Functions to Product of any Number of other Functions. It is now apparent that all the results of §§ 86-91, inclusive, may be combined into the following single more general formula :-if

where L, M, N, P, Q, R, etc., are functions of x, and if X', L', P', etc., be the x-gradients; then

etc.;

93. Theory of Resultant Error.-If the last formula be multiplied by dx, and if we call X'dx = 8X, L'dx = SL, P'dx= SP, all direct reference to x disappears and the formula becomes

Now, if L, M, etc., represent measurements of physical quantities that have been made in order to calculate from them the quotient X; and if, by any means, it be known or estimated that the measurement of L has been subject to a small error SL, and that the measurement of M has been subject to a small error SM, etc., etc.; SL then is the ratio of error in the measurement of L, and L

SM

is

M

the ratio of error in the measurement of M, and similarly with is the resulting ratio of error in the

the other factors; while

8X
X

calculated quotient X.

The above proposition may then be expressed in the following words:

The ratio of the product of a number of measured quantities to the product of another set of measured quantities is subject to a ratio of error, equal to the sum of the ratios of error in the individual multipliers reduced by the sum of the ratios of error in the individual divisors.

Attention must here be paid to the signs of the errors. Thus, if SL is a negative error, then is really a negative fraction.

SL

L

Now, although there is often reason for supposing the suspected error to lie more probably in one direction than in the opposite, still errors being things which are avoided as far as possible (or convenient, or profitable), and, therefore, not consciously incurred, it is never known for certain whether any individual error be + or. Thus, although divisors give by the formula ratios of error of opposite sign to those given by multipliers, it may happen that SX all the terms in are really of the same sign, and have, therefore,

X

all to be arithmetically added to get the whole ratio of error. Thus, if we are considering what may be the maximum possible error in X, we must pay no attention to the difference between multipliers and divisors, but add all the ratios of error in all the factors (both multipliers and divisors) together independently of sign.

It need hardly be pointed out that this maximum possible error is greater than the probable error.

If b

94. Exponential Function.—In fig. 20 is drawn a curve with horizontal ordinates called 7, and vertical ordinates b'. 7 varies continuously. Here is essentially a number; not, of course, necessarily a whole number, since its variation is continuous. b is a constant, and receives the name of the "base" of this curve. were any physical quantity, then its different powers would have various physical "dimensions," and would mean physical quantities of different kinds. In fig. 20 we assume the various vertical ordinates as all of the same kind, and therefore b cannot be a physical quantity, but must be a pure number. On this assump

tion we find that ' 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 kb’ 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 may write

b1 = bn bl-n)

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 l.

Considering various l's and various pairs of points with ordinates 7 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.

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 7 does not depend in any way upon the length n. For any one point 7, we may take various lengths n, thus getting various points (ln), 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 b' 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.