Therefore, since 1/sin l=cosecl, and 1/cos l=sec 1,

86. Reciprocal of a Function. The second and the last of these illustrations are special cases of the semi-special semi-general case-a very important one

Dividing by A, that is, multiplying by L, we obtain this in the more symmetrical form

1 αλ λαι

L'
I'

87. Product of two Functions.-Retaining the notation of the last two articles, one particular function, to which we may apply the rule of § 84, is the product XL. Thus

If X, E and x be used for the three mutually dependent variables, instead of the letters λ, L and 7; and if X' and 'be the x-gradients of X and E; then the above is written

This extremely useful result may be easily proved directly by the

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, A, 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

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 and of x is found with equal ease. Call this reciprocal X and its x-gradient

X'. Then

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 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
M

is

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

the other factors; while

SX
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,

SL

if SL is a negative error, then is really a negative fraction.

Similarly, if dP is a negative error, then (-) is really a positive

fraction.

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 8X 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.

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. If b 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