which differs from that for (x+a)" in the signs of the alternate terms, beginning with the second, which are all negative: if we replace a by x, and x by a, we shall also get

cients of

the even

it follows, therefore, that the difference of the sums of the nu- terms. merical coefficients of the odd and even terms is equal to zero, and that they are consequently equal to each other*.

of the

492. The following are examples of the application of the Examples binomial theorem to the formation of the powers of binomials.

(1) (x + a)3 = x3 + 3 ax2 + 3 a3 x + a3.

(2) (x+a)* = x* + 4ax3 + 6a2x2 + 4a3x + a*.

(3) (x+a)3 = x2 + 5 ax1 + 10 a2x2 + 10a3x2 + 5 a1x + a3.

(4) (a + x)® = a® + 6a3 x + 15 a*x2 + 20 a3x3 + 15a2 x1

+6ax 5 + x 6.

(5) (a− x)2 = a1 — 7 a® x + 21 a3 x2 – 35a*x2 + 35a3x“

- 21 a2x2 + 7a x® — x2.

(6) (a− x)1o = a1o-10a3x+45a2x2-120 a2x2+210aox1-252 a3.x3 +210 a*x* - 120 a3x2 + 45a2x2 - 10ax2 + x1o.

formation of powers of binomials.

The sum of the coefficients of the even terms will express the number of the odd combinations of n things, taken one and one, three and three, five and five, and so on together: whilst the sum of the coefficients of the odd terms, omitting the first, will express the number of the even combinations of n things taken two and two, four and four, six and six, and so on, together: it follows, therefore, that the number of odd combinations of any number of things will exceed the corresponding number of even combinations, by the first coefficient of the series or by unity.

In this example, the terms of the binomial or a2 and as may be replaced in the first instance by any two other symbols, such as c and d, and when the series for (c-d) is formed, we replace c and d by a' and ax respectively in all its terms.

+ 70x3 y1 – 168x2y3 + 224xy® − 128y'.

We shall have occasion, in Symbolical Algebra, to resume at some length the consideration of the binomial theorem when the index is not a whole number and when the series itself is indefinitely continued.

CHAPTER IX.

ON THE SOLUTION, IN WHOLE NUMBERS, OF INDETERMINATE
EQUATIONS OF THE FIRST Degree.

493.

nate equa

AN equation is indeterminate, as we have already seen Indetermi(Art. 405. Note), when it involves more unknown symbols than tions. one: and a system of simultaneous equations is reducible to a single indeterminate equation, when the number of unknown symbols exceeds the number of equations which involve them (Art. 402): under such circumstances, we can only express explicitly one unknown symbol in terms of the others, whose value is therefore dependent upon them.

494. Thus the equation

3x + 7y = 61

becomes, when solved with respect to x, (Art. 405. Note), ·

and it is obvious that if any value whatever be assigned to y between 0 and

61

7,

which are its limits, we shall be enabled

to determine corresponding and possible values of x: thus if

Corresponding values of the unknown symbols in an indeterminate

equation.

495. If, however, the conditions of the problem propose had been such as to restrict the use of the word number (Art. 416) to mean whole numbers only, excluding fractions altogether, w should find only three pairs of values of x and y, which would answer the requisite conditions: these are

Problems in which the word tember means whole

numbers only.

496. There are many problems, leading to indeterminate equations, in which the values of the unknown symbols, which they involve, are thus restricted to denote whole numbers only. of this kind are the following.

(1) To find a number such, that five times the first digit added to six times the second shall be equal to 50.

If x and y be the two digits, the conditions of the problem lead immediately to the equation

5x+6y= 50.

There is only one pair of integral values or digits, namely, x = 4 and y = 5, which satisfy this equation, which ceases therefore to be indeterminate when viewed with reference to the problem proposed.

(2) To find a number, which divided by 7 shall leave a remainder 6 and which divided by 10 shall leave a remainder 7.

If we call

by 7, and y its

further denote the

and therefore

the quotient of this number when divided quotient when divided by 10, and if we number itself by n, we shall get

n =7x+6=10y + 7,

7x-10y=1:

is obvious that x and y, or the quotients of the divisions, e necessarily whole numbers.

The successive pairs of integral values of x and y, and e corresponding values of n, will be found to be as follows:

We thus find that the values of x increase by 10 (the oefficient of y), and the values of y increase by 7 (the cofficient of n), and that the series of them is unlimited: the corresponding series of numbers 27, 97, 167, 237, 307, &c. will be severally found to answer the conditions of the problem.

There are

only two

497. There are only two different forms of indeterminate equations of the first degree, when completely reduced, (Art. forms of 394.), which are

sup

where a, b and c are whole numbers: we may further
pose that a, b and c have no common measure: for if such
a common measure exists, we may simplify the equation by
dividing all its terms by it: if, however, a be not prime to
b, in the equation thus reduced, its solution will be impossible
in whole numbers: for since a and b, x and y are whole
numbers, c, which is either the sum or difference of ar and
by, must be divisible by any common measure of a and b,
(Art. 101.), which is contrary to the hypothesis which has been
made.

indeterminate equations of the first degree which involve two

unknown

symbols.

A particular solution of the

498. A particular solution of either of the equations, (Art. equation 196.),

or

......

ax-by=1. .(1),
by-ax = 1.

...(2),.

ar-by=1

or

by-ax=1, will lead to

the general solution of

will immediately lead to the general solution of the equations the equa

ax-by= c... .(3),

and ax + by = c... ·(4).

tions
ax-by=c

or

ax+by=c.