An equivalent series

should pos

sess as many values, if more than one, as the

expression in which it

originates.

sign of the symbol or combination of symbols according to which it is arranged, we should at once conclude that its odd powers alone presented themselves in one series, and its even powers in the other: thus in assuming series, proceeding according to powers of x for sin x and cos x, we should on this account, exclude all terms involving the even powers of x from one series, and all terms involving its odd powers from the other.

959. A series, in order to be completely equivalent to the expression from which it is derived, should possess the same number of values with it, when those values are more than one: but the process of developement will in many cases apply exclusively, in the first instance, to the deduction of that form of the series which represents its arithmetical value, and which is unique, leaving the other values to be supplied by various methods, some of which will be explained hereafter: thus the series which the binomial theorem, as commonly applied, gives for (1 + x) is

which expresses the arithmetical value of the biquadratic root of 1 + x, when xa is less than 1: its complete developement would be expressed by

where 1 is the recipient (Art. 724) of the multiple values of the biquadratic root, which may also be replaced by cos + -1

2rT

4

Whatever value a

radical or

where the value of the series, included between the brackets, is unique, but that of (ax), which is multiplied into it, is multiple: the successive substitution of the three values, which (ax)} admits of, will give the triple series which represents the complete developement.

960. But if a radical or other term, admitting of multiple values, presents itself in a generating expression or function, and

pression,

subsequently reappears in the equivalent series into which it is other exdeveloped, it should be kept in mind that, whatever be the value admitting of multiple which it is assumed to possess on one side of the sign of equality, values, posit must continue to retain it on the other: thus, in the series

whichever of its two values, Jr, whether + or

1

to possess in
the same is retained throughout the series:
1 + √x'
it is only, when such radical or other expressions (such as equi-
sinal and equicosinal angles) present themselves in equations,
whose members are neither identical nor reducible, by the per-
formance of the operations indicated, to identity, and where no
convention expressed or understood limits the multiplicity of
their values, that we admit every combination of such values
as being equally possible, and as being equally included within
the range of the different cases to be considered.

The consideration of this subject, which is one of the most important in the theory of series, will be resumed in a subsequent Chapter.

sesses in
one term of
a series, it
retains in

all.

of series.

961. When an expression, denoted by a symbol y, is de- Inversion veloped in a series proceeding according to powers of any other symbol or quantity x, which it involves and upon which its value is dependent, we may invert the operation, and express x or the symbol according to which the first series is arranged, by means of another series, proceeding according to powers of the symbol y: thus if

and the problem proposed for solution is the following: "Given the first series or its successive coefficients A1, A2, A ̧....... to find the second series or its successive coefficients a,, a,, Az...." We shall exemplify it in the determination of the series for the measure of an angle in terms of its sine, the series for the sine of an angle in terms of its measure being given.

for it is obvious that this series (2) must be confined to odd powers of sin x, for the same reason that the series (1) for sin a was confined to odd powers of x: replacing in series (1), ≈ and its powers by the series assumed to express it in (2), we get x = А。 sin x + A ̧ sin3 x + А, sin3 x +

...

A sin x

Consequently, adding together the terms on both sides, we get

and equating the coefficients of corresponding terms, we find

The law of formation of the coefficients of this series is not made manifest by the number of its terms which have been thus determined, and it appears that the extension of the process to other terms would involve the formation of the successive powers of

A, sin A, sin3 x + A, sin x + .......

and requiring the aid of a theorem, called the multinomial
theorem, which we have not hitherto investigated: in the pro-
gress of
our investigations, however, we shall find other and more
expeditious methods of effecting this and other developements,
in which the laborious formation of complicated powers and
products will be altogether avoided.

series for

962. It has been shewn (Art. 778) that all measures of angles The inverse included in the expressions 2r+x and (2r+1) π-x are equi- sin x is only sinal, and we may conclude that all such values of x are equally true for the included in the equation

least of the measures of the equisinal angles.

(1):

but it is the least of these equisinal values of x which is alone admissible in the inverse series

which is necessarily convergent and arithmetical in its value. This observation will be found hereafter to apply to inverse series and expressions generally, and is connected with important theories.

CHAPTER XLI.

The extrac

ple and

cube roots

ON THE SOLUTION AND THEORY OF CUBIC EQUATIONS.

963. WE have given, in a former Chapter, rules for the tion of sim- extraction of simple and compound cube roots (Art. 242), as compound far as those rules could be properly considered as included is the solu- within the province of arithmetic: the processes in question are equivalent to the arithmetical solution of binomial and other cubic equations, in cases where they necessarily possess an arithmetical root, whose determination involves no ambiguity: we propose, in the present Chapter, to consider the general theory of the solution of cubic equations, and to exemplify the arithmetical or other rules to which it leads.

tion of a cubic equation.

The second term of a complete

964. A cubic equation, cleared of fractions and radical expressions, by the rules given in Chap. V., may be always cubic equa- reduced to the form

tion may always be obliterated.

where a, b, c are whole or fractional numbers, positive or negative: but this equation may be farther reduced by a very simple process, to a form in which its second term will disappear: for, if we make x =

a 3

y+ we get