was to the whole price of the worse as 72 to 7. How many yards had he of each?

Ans. 9 yards of the better, and 7 of the worse. Prob. 27. There are four towns in the order of the letters, A, B, C, D. The difference between the distances, from A to B, and from B to C, is greater by four miles than the distance from B to D. Also the number of miles between B and D is equal to two-thirds of the number between A to C. And the number between A and B is to the number between C and D as seven times the number between B and C: 26. Required the respective distances.

Ans. AB=42, BC=6, and CD=26 miles.

CHAPTER XII.

ON

THE BINOMIAL THEOREM.

434. Previous to the investigation of the Binomiat Theorem, it is necessary to observe, that any two algebraic expressions are said to be identical, when they are of the same value, for all values of the letters of which they are composed. Thus, x-1=-1, is an identical equation: and shows that x is indeterminate; or that the equation will be satisfied by substituting, for x, any quantity whatever.

Also, (x+a)x(x-a) and x2-a2, are identical expressions; that is, (x+a)x(x—a)=x3—a2; whatever numeral values may be given to the quantities represented by x and a.

435. When the two members of any identity consist of the same successive powers of some indefinite quantity x, the coefficient of all the like powers of x, in that identity, will be equal to each other.

For, let the proposed identity consist of an indefnite number of terms, as,

a+bx+cx2+dx3 + &c. = a+b'x+c2x2+d'x3 + &c. Then, since it will hold good, whatever may be the value of x, let x=0, and we shall have, from the vanishing of the rest of the terms, a=a'.

Whence, suppressing these two terms, as being equal to each other, there will arise the new identity bx+cx2+dx3+ &c. = b'+c'x2+dx3 + &c. which, by dividing each of its terms by x, becomes

b+cx+dx2+ &c. = b'+c'x+d'x2+ &c.

And, consequently, if this be treated in the same manner as the former, by taking x=0, we shall have b=b', and so on; the same mode of reasoning giving c=c', d=d', &c., as was to be shown.

§ I. INVESTIGATION OF THE BINOMIAL THEOREM.

436. NEWTON, as is well known, left no demonstration of this celebrated theorem, but appears, as has already been observed, (Art. 163), to have deduced it merely from an induction of particular cases, and though no doubt can be entertained of its truth from its having been found to succeed in all the instances in which it has been applied, yet, agreeably to the rigour that ought to be observed in the establishment of every mathematical theory, and especially in a fundamental proposition of such general use and application, it is necessary that as regular and strict a proof should be given of it as the nature of the subject, and the state of analysis will admit.

437. In order to avoid entering into a too prolix investigation of the simple and well-known elements, upon which the general formula depends, it will be sufficient to observe, that it can be easily shown, from some of the first and most common rules of Algebra, that whatever may be the operations which the index (m) directs to be performed upon the expression (a+x), whether of elevation, division, or extraction of roots, the terms of the resulting series will necessarily arise, by the regular integral powers of x; and that the first two terms of this series will

always be a+max; so that the entire expansion of it may be represented under the form

am+ma1x+BaTM-2x2+CaTM+DaTM••• x3+&c.

Where B, C, D, &c. are certain numerical coefficients, that are independent of the values of a and x; which two latter may be considered as denoting any quantities whatever.

438. For supposing the index m to be an integer, and taking a=1, which will render the following part of the investigation more simple, and equally answer the purpose intended; it is plain that we shall have, according to what has been shown (Art. 289), (1+x)=1+mx+bx2 +cx2+dx++ &c.

(1). 439. And if the index m, of the given binomial,

be negative, it will be found by division, that (1+x), or the equivalent expression

腐

(1+x) &c.

1

-—1—mx-b'x2 - c'x3

I+mx+bx2+cx3 &c.

2

where the law of the terms, in each of these cases is similar to that above mentioned.

440. Again, let there be taken the binomial

m

n

(1+r), having the fractional index ; where m and n are whole positive numbers.

Then, since (1+x) is the nth power (1+x)~ and, as above shown, (1+x)=1+ax+b2+cx3+dx

+ &c., such a series must be assumed for (1+x) »' that, when raised to the nth power, will give a series of the form 1+ax+bx2 +cx3+dx2+ &c.

But the nth or any other integral power of the series 1+px+qx2+rx3+sx2+ &c. will be found, by actual multiplication, to give a series of the form here mentioned; whence, in this case, also, it necessarily follows, that

(1+x)=1+px+qx2 + rx3+sx++ &c.

And if each side of this last expression be raised to the nth power, we shall have (1+x)TM=[1+(px+qx2 +rx3+sx*+ &c.)]"; or, by actual involution,

3

1+mx+bx2+ cx3+ &c. = 1+n (px+qx2 + &c.) +&c.

Whence, by comparing the coefficients of x, on each side of this last equation, we shall have, accord

ing to (Art. 435), np=m, or p=

case,

m

n

m

n

; so that, in this

(: +x)=1+x+qx2+rx2+sxa +&c. ・・・ (2) ; where the coefficient of the second term, and the several powers of x, follow the same law as in the case of integral powers.

m

441. Lastly, if the index be negative, it will be

n

m

found by division as above, that (1+x)−☎ equivalent expression,

or the

where the series still follows the same law as before. 442. And as the several cases, (1, 2, 3), here given, are of the same kind with those that are designed to be expressed in universal terms, by the general formula; it is in vain, as far as regards the first two terms, and the general form of the series, to look for any other origin of them than what may be derived from these, or other similar operations.

443. Hence, because (a+x)"=a" (1+2)", if

there be assumed (a+x)"=am+maTM--1x+Bx2+Cx3 Dx &c.; or which will be more commodious, and equally answer the design proposed,

it will only remain to determine the values of the coefficients A,, A., As, &c. and to show the law of their dependence on the index (m) of the operation by which they are produced.

444. For this purpose, let m denote any number whatever, whole or fractional, positive or negative; and for in the above formula, put y+z; then,

α

there will arise (1+2)==[1+(y+z)]"=[(1+y)+

z]m, which being all identical expressions, their expansions, when taken according to the above form, will evidently be equal to each other.

445. Whence, as the numeral coefficients A1, A2, A., &c, of the developed formulæ, will not change for any value that can be given to a and x, provided the index (m), remains the same, the two latter may be exhibited under the forms

[!+(1+z)]m=1+A,(y + z)+A2(y+z)2+ &c. [(1+y)+z]=(1+y)m+A, z(1+y)-'+A2z2 (1+y) m--2+ &c.

And, consequently, by raising the several terms of the first of these series to their proper powers, and putting 1+y=p in the latter, we shall shall have i+A, (y+z) +A, (y2+2yz+z3) +A ̧( y3+3y2 z+ 3yz2+z3)+&c.=pm+A ̧pm-1z+A2pm-2z3 +A ̧pm-dz 3

+ &c.

446. Or, by ordering the terms, so that those which are affected with the same power of z may be all brought together, and arranged under the same head, this last expression will stand thus: z+A2 z2+A, A1y+2A,y +3Agy +4Ay A,y2+3A3y2 +6A4y2+10Ã ̧ya

1+A,

23+ &c. (5).

+30Ay

&c.

=pm+A‚pm-12+A2pTM¬ +A3pm¬31⁄23+ &c.