Prob. 21, Bacchus caught Silenus asleep by the side of a full cask, and seized the opportunity of drinking, which he continued for two-thirds of the time that Silenus would have taken to empty the whole cask. After that Silenus awoke, and drank what Bacchus had left. Had they drunk both together, it would have been emptied two hours sooner, and Bacchus would have drunk only half what he left Silenus. Required the time in which they could have emptied the cask separately.

Ans. Silenus in 3 hours, and Bacchus in e. Prob. 22. Two persons. A and B, talking of their money, A says to B, if I had as many dollars at 5s. 6d. each, as I have shillings, I should have as much money as you; but, if the number of my shillings were squared, I should have twice as much as you, and 12 shillings more. What had each?

Ans. A had 12, and B 66 shillings. Prob. 23. It is required to find two numbers, such, that if their product be added to their sum it shall make 62; and if their suin be taken from the sum of their squares it shall leave 86.

Ans. 8, and 6. Prob. 24. It is required to find two numbers, such, that their difference shall be 98, and the difference of their cube roots 2.

Ans. 125, and 27. Prob. 25. There is a number consisting of two digits. The left-hand digit is equal to 3 times the right-hand digit; and if 12 be subtracted from the number itself, the remainder will be equal to the square of the left-hand digit. What is the number? Aus. 93.

Prob. 26. A person bought a quantity of cloth of two sorts for 77. 18 shillings. For every yard of the better sort he gave as many shillings as he had yards in all; and for every yard of the worse as many shillings as there were yards of the better sort more than of the worse. And the whole price of the better sort

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 Required the respective distances.

C: 26.

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

CHAPTER XII.

ON

THE BINOMIAL THEOREM.

434. Previous to the investigation of the Binomial 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—a)=x2—aa; 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 indefinite number of terms, as,

3

a+bx + cx2+dx3 + &c. = a + bx + c2x2 + dx3 + &c. a+bx+cx2+dx3+ 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, suppressiug these two terms, as being equal to each other, there will arise the new identity bx + cx2 + dx3 + &c. = b'+c2x2 + dx3 + &c. which, by dividing each of its terms by x, becomes

3

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

And, consequently, if this be treated in the same manner as the former, by taking x=0, we shall have bb', 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+male; so that the entire expansion of it may be represented under the form

am+maTM-x+Bam-2x2 + Cam¬3+Dam--4x3 +&c. Where B, C, D, &c. are certain numerical coefficients, that are independent of the values of a and ; 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 +cx3+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

1

(1+x)

&c.

1

1+mx+bx2+cx3 &c.

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

(1+x), having the fractional index ;

and n are whole positive numbers.

m

n

where m

Then, since (1+x) is the nth power (1+x); and, as above shown, (1+x)=1+ax+b2 +çx3+dx1

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

But the nth or any other integral power of the series 1+px+qr2 + 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

ገቢ

(1 + x) *=1+px+qx2 +ræ3 +sx2 + &c.

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

m

m

n

; so that, in this

(1+x)=1+=x+qx2+rx2+sxa+ &c. ... (2);

n

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) or the equivalent expression,

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.

if

443. Hence, because (a+x)"=a"(1+~)", it

=

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