be converted into a series of single terms. And in like manner, if for z we write b + v, we obtain the expansion of (x + a + b + v), and so on. In art. [276] we found the general term of the expansion of the binomial (x+a)m, we may thus find the same for the polynomial (a + b + c + &c. + h)m, m being considered a whole number. We have seen, art. [276], that in this case the coefficient of am-" x" in (a+x)m was 1 2 .... n m (m-1)...(m-n + 1) , and that this was also the coefficient of an xm-n Call this coefficient M. In the above polynomial expression suppose b + c + &c. + g + h = x. It becomes (a+x)m, and the general term of this expansion is Ma" xm-", that is, giving n all successive values between 0 and n we obtain each particular term. Call m -n, m1; the general term becomes M a" a1, and we see that n and m1 may have any integral and positive values subject to the condition that n + m1 = m. Again, since b + c + &c. + h = x. Suppose we have and Let M1 = c + &c. + h = y, x = b + y, xm1 = (b + y)1. m1 (m-1)...(m1-p+I) 293. The same theorem enables us to find any power of a polynomial expression. Thus having (x+y)m =xm+mxm-1 y +m m-1 2 11 xm-2 y2 + &c., for y write a + z, and this becomes xm-2 (a+z)2+&c., expanding the terms (a + z), (a + z)3, &c. by the binomial theorem, this may * The series for the binomial may be represented in other forms which may sometimes be used with 1.2 a-x 2 x a+ a+za+x 1 a-x 1a+x 2x ...a+ = 1 a-x a + x 1 1 2 (-)* n.n+1 a + x a-x a + x + &c.). 2 + &c. Then the general term of the last expan sion is M, brym ym-p, or M1 br yma, ma, being equal to m1 m p, that is to n-p, or m + n + p being = m. Hence the general term of (a + n), or (a + b + y) is M M1 a" brys, the only condition of the 3 exponents, which must, of course, be all positive, being that their sum = m. Then making y = c + z, and proceeding as before, and so on successively, it is clear that we shall at last arrive at the following term as the general term of the expansion of (a + b + &c, + g + h)m MM, M... M. an bp cq.. g "hmi, the little letter (i) not representing an exponent, but the subscript number to the letter m to which it is affixed. As before, the law of the exponents is, that, being all positive, their sum = m. : Now M = M1 = M2 = 2 This quantity will represent every term of the developement by giving to np...wall the values of which they are capable, subject to the above condition. Observe that the condition being fulfilled any of them may be equal to zero, in which case the sets of factors 1.2...n, 1.2... p, &c. belonging to those which are zero must be omitted altogether. Of Interest. 294. Interest is the value of the use of money. This value depends upon the plenty or scarcity of unemployed capital in the country, the rate of profits, and various considerations of the same nature, all, however, following from, and comprised in, the first. Estimated in this way, the value of the use of a given sum of money, though it would be a difficult problem to assign it, is m1 (m - 1).. (m1 - p + 1) evidently, the circumstances of the coun m (m - 1).. (m - n + 1) 1.2 1 2 .. .. n P &c. = &c. .. And the last factor will evidently be و try remaining the same, a fixed quantity. But the consideration paid for the use of money, which is what most writers have defined interest to be, is always a matter of previous agreement between the parties to the transaction, M1 = m; (m-1)).. (m; - v + 1) and though necessarily dependent upon its value as above estimated is yet influenced by other reasons; -the lender considering the probability that the money will be returned, and the interest regularly paid, and the borrower regarding his own circumstances, and the improvement of his expectations and opportunities by the command of a larger capital. 295. The method of settling the consideration to be paid for the use of any sum of money during any time is, by declaring that which is to be paid at the end of a certain time for the use of a certain sum during that time. This section will be employed in showing how the first is derived from the second. The time made use of in the present day is ordinarily a year, and the sum £100.; £5., or any sum less than £5., (the laws at present forbid a larger,) may be agreed upon and enforced as the consideration, or interest. In the case of £5. being agreed upon, the scale of remuneration, or as it is usually called the rate of interest, is 5 per centum per annum, or more shortly 5 per cent. Similarly had £4. been agreed upon the rate would have been 4 per cent. We might evi dently have referred the rate of interest to any other period besides a year. The Greeks and Romans referred it to a month, and made interest payable (and wisely, for the reader will collect from what follows, that a short interval is desirable) monthly; and, indeed, with us, though the rate of interest is fixed by the sum paid for the use during a year, that sum is usually made payable in 2 equal parts half-yearly, and sometimes in 4 equal parts quarterly.. We may here remark, that the sum upon which interest is charged is called the principal sum, or more shortly the principal. 296. Interest is either simple or compound, according to the manner in which it is calculated. Any sum being due or lent, at the end of a certain time, a year for instance, the interest upon it becomes payable, so that the sum then due, instead of being the sum originally lent, is that sum increased by the interest for a year. If the whole be still unpaid, and interest be still charged upon that sum only which was originally lent, and so on continually after any number of years, then the money is said to be charged with simple interest. In this case it is clear, that the amount of interest is in proportion to the time. But if after the first year, when interest is payable and unpaid, the principal sum and interest due upon it be considered as a new principal sum, and charged with interest accordingly, and so on continually, then the money is said to be charged with compound interest. To illustrate this briefly; £100. is due from B to A, and until payment is to be charged with 5 per cent. simple interest. At the end of the first year £5. is due for interest, at the end of the second year £5. more, and so on £5. for each year, so that the whole interest for any number of years is equal to £5. multiplied by that number. But if the £100. be charged with compound interest, at the end of the first year £5. is due for interest, but at the end of the second year not £5. more, which is the interest of £100. for a year, but a larger sum, namely the interest of £105. at the same rate, and so on. terest is 297. The calculation of simple insufficiently easy. From the interest of a hundred pounds for a year we can by a simple proportion find the interest of any other sum for the same time, and knowing the interest for a would, at the given rate of interest, amount to the debt at the time when it is due. The discount is the difference between the debt and the present value. Now we know immediately, from the rate per cent., what £100. would amount to in the given time. And the present value would in the same time amount to the given debt. Hence the present value of a sum of money due in a given time is the fourth term of a proportion, the three first terms of which are the numbers representing in pounds the amount of £100. in the given time, 100, and the given sum. Hence we have the following rule: Multiply the given sum by 100, and divide by the number representing the amount of £100. in the given time. What is the discount on £36. 10s. due in 3 months, the rate of interest being 4 per cent. ? £100. in 3 months amounts to £101. M = P+Pnr = P(1 + nr). (2.) The equation (1) is, under a slightly different form, the rule given in art. [292]. There being four different quantities in each of the above equations, by knowing any three of them we can obtain the other. We shall find it necessary in applying the equations to reduce shillings, &c. to decimals of a pound, and months, &c. to decimals of a year. Required the interest on £14. 58. for a year and a half, at 5 per cent. By equation (1) I = Pnr, n = 14 years = 1.5, Multiplying in the manner adopted in art. [167]. It is unnecessary to give examples of these expressions, as they are of the same nature as those of the last article. 301. In almost all money transactions, it is usual, when a deduction is made by way of discount in consequence of immediate payment, to calculate the interest of the sum to be paid, instead of the discount as above given. This gives an advantage to the person so paying, inasmuch as he deducts the interest of the sum to be paid instead of the interest of its present value. But the person receiving is is w willing to forfeit the difference for being freed from all doubts and uncertainty. In the same way interest is substituted for discount in the general method of calculating equations of payments. A owes B £P, due at the end of ni years, and £P2, due at the end of n2 years from the present time; at what time must he pay B the sum of £P1 and P2, that neither party may gain or lose? Let n be the number of years required. Then (n - n1) years is the 1 which expression is tantamount to the rule usually given: Add together the products of each debt multiplied by the time when it is due, and divide by the sum of the debts. Here, as before, the substitution of interest for discount is to the advantage of the debtor. The rule is so simple that it is unnecessary to illustrate it by examples. 302. As soon as a sum of money is payable, it matters little whether it be due under the name of principal or interest; the use of it would be of equal value to its owner. It would, therefore, appear to be equitable that it should be charged with interest in one case as well as the other; in other words, that a debt forborne should be charged with compound interest. It is, however, a singular fact that the laws of H |