9. What is the difference between the value of a freehold estate, or perpetual annuity of £100 per annum, and that of a leasehold estate of £100 per annum, to continue 60 years? Ans. The freehold is worth £107 1s. 4 d. more than the leasehold. 10. What sum ought to be paid for the reversion of an annuity of £50 for 14 years after the next 7, that the purchaser may make 5 per cent. of his money? Ans. £351 10s. nearly. 11. The sum of £518 6s. being placed out at compound interest for 3 years, amounts to £600; find the rate of inAns. 5 per cent. terest. 12. In what time will a sum of money double itself, at 4 per cent. compound interest? Ans. In 17.6 years. 13. Suppose a person to place out annually the sum of £20 for 40 successive years, what would the whole amount to at the end of that time, at 5 per cent. compound interest? Ans. £2416. 14. Find the present value of an annuity of £40, to continue 5 years, allowing compound interest at 5 per cent. Ans. £173 3s. 7d. 15. What must be paid for an estate whose annual rental is £79 4s. that the purchaser may make 4 per cent. of his money? Ans. £1760. 16. A person places P pounds at interest, and adds to his capital at the end of every year th part of the interest for that year; what is the amount at the expiration of n years. 1 m Ans. P ( r m +r+ 1). 17. Find the present value of an annuity of £20, to commence 10 years hence, and then to continue 11 years, allowing 3 per cent. compound interest. Ans. £118 7s. 3d. 18. A person leaves to his two sons, A and B, equal shares of an estate producing £1000 per annum, but A proposes that B shall take both shares, and allow him an equivalent annuity for 20 years; what annuity ought Pallow A, reckoning interest at 41 per cent." Ans. £854 2s. 4d APPENDIX, CONTAINING MISCELLANEOUS INVESTIGATIONS AND EXAMPLES. I. In the equation ar" + br2-1 + cr”-2 + .............. pr2 + qr + 8 = N, r and N are given numbers: it is required to find the numerical values of the coefficients a, b, c, &c., and also the value of the exponent n. Divide the second side of the equation by the known number r; the quotient will be and the remainder s, which thus becomes known. In like manner, dividing the number Q by r, the quotient is ann-2 tởm”-3 teen-t p = Q', ...... and the remainder q. And it is plain, that by thus dividing N, Q, Q', &c., by r, the successive remainders will be the required coefficients s, q, p, &c.; and the number of divisions, minus 1, will denote the value of n. These coefficients will be no other than the digits or figures which express any number N in the arithmetical scale of notation in which the radix is r; the radix in the common or decimal scale is 10. For example, suppose it were required to convert the number 17486 from the decimal to the senary scale, that is, to express it in the scale of which the radix is 6, the range of figures in this scale being 0, 1, 2, 3, 4, 5. Here we have to find the coefficients a, b, c, &c., so as to fulfil the condition 1-2 ...... s = 17486. a. 6" + b. 6”—1 + c . 6”—2 + Dividing therefore successively by 6, and noting the several remainders, we have Therefore 17486, in the decimal, or denary scale, and which means 1.10 + 7. 103 + 4. 102 + 8.10 + 6, when converted into the senary scale is 212542, which means 2.6 +1.6 + 2 . 63 + 5 . 62 + 4 . 6 + 2 = 17486. In the common or denary scale every digit is increased in a tenfold proportion by being advanced a place to the left; in the senary scale the advance increases the digit in a sixfold proportion; in the quaternary scale in a fourfold proportion; in the ternary, in a threefold; and in the binary, in a twofold. For ample information on scales of notation the learner 'may consult Barlow's "Theory of Numbers." II. To find the value of a vanishing fraction, that is, of a fraction the terms of which vanish for a particular value of some general symbol common to both. Р Let be a fraction such that a certain value a being put Q for x (a variable quantity supposed here to enter P and Q), it is found that, for such value of x, there results P = 0 0, and Q = 0, the fraction then taking the form : it is required to 0 find what the true value really is which is concealed under this ambiguous form. It is plain, that in order that P and Q may simultaneously Р vanish for xa, the fraction must be of the form p(x-a)m 9(x-a)"' fractional; P and Q could not vanish together for any value of x, unless for that value there vanished some factor common to P and Q. If therefore this common factor be cancelled, the exponents m, n, one or both, being integral or 0 0 the cause of the vague form will be removed, and the true value of the fraction obtained, as in the following examples: For this value of x, the fraction takes the form sign that the numerator and denominator have a common measure that also vanishes for x = a. It is easy to see that a is this common measure, and that X In order to put in evidence the vanishing factor, it will be convenient to write the fraction thus ✓ { (x − a) (x − a)} + √ { 2a (x − a) } √ { (x + a) (x − a) } }, in which form it is at once seen that the vanishing factor common to numerator and denominator is (-a). Cancelling this common factor, the general fraction becomes √(x − a) + √ 2a √ (x + a) and the particular case of it for x = a is evidently = 1. √2a Here the vanishing factor, common to numerator and denominator, must evidently be 1 x, at least; probably (1-x)'. 3x 2x+2x3 The quotient I x2 + x 1 is clearly divisible also by x, the second quotient being 1 + x + x. Consequently By the Binomial Theorem, the numerator of this fraction therefore dividing by x, the denominator, the general value of the fraction is III. To determine the greatest or least values which algebraical expressions admit of under proposed conditions. To find what particular value a variable quantity must have in order that the expression into which it enters may be greater or less than it would be for any other value if that variable is a problem belonging to what is called the maxima and minima of algebraical quantities, and which in its widest extent can be solved only by help of the differential calculus (see the Rudimentary Treatise on that subject). But many questions belonging to maxima and minima can be solved by |