NOTATION.

Letters near the beginning of the alphabet denote constants which may in general be positive or negative, whole or fractional, real quantities or numbers. Those near the end of the alphabet denote variables.

The symbol stands for "denotes " or "is identical with." The symbols >, », <, ✶ stand for greater than, not greater than, less than, not less than, respectively.

The symbols f( ), F( ), ( ), 4(), denote any function of the quantity placed inside the brackets, except when restricted by the

context.

X, E, X, x are briefer symbols for functions of x.

L, M, N, P, Q, R also sometimes denote functions of the variable. Y any function of the variable y which is independent of x. f(a)=the same function of a that f(x) is of the variable x.

=

[ƒ(x)] =ƒ(b) − f(a) ƒ'(x)=df(x). ƒ"(x)=1}f(x)

:

dx

dx2

dX

d2X

dX

: X"=

X (n) ==

dx

dx2

dxn

dY

dy

ƒ'(x)=dƒ(X): ƒ'(Y)='

dY

a

a=partial differentiation with respect to x.

ƒ(x,y)=the_differential coefficient with respect to x of the function f(x,y) where x and y are independent variables.

ƒ (x,y)=ƒ (x,y)=the second differential coefficient of f(x,y) with

respect to x and y, where x and y are independent variables.

[Xdz=Integral of X with respect to x
["Xde=[[Xde=[[Xdede=[ { [Xdx} dz

(2)

(n)

||"X" = || [[f(x,y)dydx=[{ [f(x,y)dy } dx

= [ ] - - (n føymdots) Xdx” = [ ] - - (n formbois) Xdxdx - - (ndx's)

Σ or ()=Sum of a series of terms of the same type as that following or placed within the brackets.

[Me)de=[[{z}de]=the limiting value towards which

[f(x)dx approaches as dx ap

[f(x)dx=[*f(x)dx

proaches the limiting value 0.

a

n! 1x2 x3.--(n-1) x n; (n being a positive integer). log10=log x in the Decimal or Common system of logarithms. log log in the Natural or Neperian

e=2.7182818

1 1 1

=1+ + + + ---- = base of Neperian 2! 3!

logarithms

B1, B2----(Bernoulli's Numbers).*

In trigonometric functions the angles are given in "circular

measure," the unit being the radian which

nearly.

0

180°

or 57° 17′ 45′′

π

e-x-1dx (the Gamma Function). See X. 1-6.

ABBREVIATIONS.

Sp. Case Special Case.

H

Exc. Exceptional Case, or Case of failure.

* See note at end of Section VIII.

Sp. Case: -1: Xdr-X-X+X-X" + etc.

x2 1.2

X'

4!

dx=2·3026 ----log1X + C = log. X+C

APPROXIMATE INTEGRATION.

-α

b − +

Xdx= @X + 2X, +2X,+---+2X,_+X,)

2n

(iii.)

b

α

-6

f(x)dx=h[} {ƒ(b) +ƒ(a)} +ƒ(a + h) +f(a+2h)

+ -- -- +f(b − h)

Boh
-ƒ (5) (x) +
6!

*See note at end of Sect. VIII.

--

11.

METHOD OF UNDETERMINED COEFFICIENTS.

(i.) If a function is known to be expressible in a certain form
containing unknown coefficients, these coefficients can
be determined by transforming the identity (see below)
and equating coefficients of like terms on either side;
like terms being those whose variable part is the same
function of the variable (e.g., the same power, or the
same trigonometrical function), or in both of which
alike the variable is absent (constant terms).
The transformation referred to may be :-

(a) Differentiating both sides (as in III. B. 18).

(b) Clearing of fractions (as in II. E. 2, etc.), or otherwise.

(ii.) Another method: Give to the primitive variable as many different values, in the identity, as there are coefficients to be determined; whereby we get as many equations as are necessary to determine them.

GENERAL NOTE re IMAGINARIES.

Some of the formulas here given contain parts which would become imaginary if the quantities involved took values outside certain limits: becoming, e.g., square roots and logarithms of negative quantities, inverse sines of quantities greater than unity, etc. When a definite integral is deduced from the indefinite one, these imaginaries, explicitly or implicitly, cancel one another, if the subject of integration is itself real. But, in many instances, two or more forms are given for the integral of the same function (e.g., III. B. 6, VI. 5, 6), of which that one is to be selected which, for the values of the constants in the particular problem under consideration, is free from imaginaries.

Some of these formulas contain parts which are imaginary for certain values of x only, whatever the constants may be, and others do so for all values of x when the constants are outside certain limits. E.g., the formula sin is imaginary when x>a, a and + a. On the other hand, the

but not when x lies between

formula

x -1

α

1

sinh -1 √b

{}

contains imaginaries when bis

negative, whatever the value of x may be.

In these classified tables, the conditions under which a formula involves imaginaries are, as a rule, pointed out in cases of the latter sort, but not in those of the former.