a smaller premium, since, commencing at the same time, it continues longer. , to continue until the receipt of the benefit is determined. Required 7. Here the benefit side is, by (I.), and (XIII.), A[D(x + n) + M(x) — M(x+n)]; and the payment side is, by (IV.), [N(x − 1 ) − N (x + n − 1)]. ... [N(x-1)-N(x+n-1)]=A[D(x+ n) + M(x) - M(x+n)]. Whence, π= If A= £100, x=30, and n=30; that is, if the benefit be receivable at 60, or at death, if before 60, we have, T= 100[D(60) + M(30) - M(60)] = 41857.53+74295.67-24598.20 39382 08-4487:42 2.6238= £2 12s. 6d. Example 5. Required the annual premium,, payable till death, for an assurance of £A, on (x), with which the whole of the premiums paid are to be returned. 91555.00 34894.66 Here, the sum to be received if death take place in the first year is A+π, if in the second, A+27, and so on. Therefore, the benefit side is, by (XIX.), (A+) M(x) + TR(x+1)=AM(x)+ R(x), by (10); and the payment side is, by (II.), πN(x−1). .'. πN(x-1)=AM(x)+πR(x); whence we obtain, N(29)-R(30) 39382 08-17127:42 Example 6.-An assurance of £A, for n years, on (a), is to be paid for by an annual premium, π, also to last years. Required . .. 74295.67 22254.65 =3.3338 £3 6s. 8d.* Here the benefit side is, by (XIII.); A [M(x)-M(x+2)], and the paymen side, by (IV.), [N(x − 1) −N(x + n − 1)]' •'• π[N(x − 1) − N(x + n − 1)] = A[M(x) − M(x+n)]; This expression assumes a very convenient form, if, instead of column M, we use its value in terms of N. Thus, since, by (13), M(x)=vN(x − 1) −N(x), and M(x + n) = vN (x + n − 1) −N(x+ n), we have, π= The Commutation Table, as it stands, does not enable us, conveniently, to find the amount of annual premium equivalent to a benefit which consists partly of a return of all the premiums paid, with simple interest upon them from the date of payment, the incremental portion being in this case of the form m, 3m, 6m, 10m, &c. The addition of another column, formed from R as R is formed from M, would however afford the means of doing so. We might obviously add as many columns as we pleased in this way. Their properties would be such that, calling the column formed from R the first, the division of any number in the nth column, by the corresponding number in D, would give the present value of an assurance whose payments should be the series of figurate numbers of the nth order; and the remark may be extended, mutatis mulandis, to the annuity columns. But such properties being more curious than useful, we do not insist upon them.-See De Morgan, I., p. 23. If A=£100, x=30, and n=10, the solution by the first of these formulæ will be, 100[M(30) – M(40)] 100(742·9567 – 522·6503) π= 23030.64 18075.46 _N(30)-N(40)=100(961539 18075 46 Example 7.-An annuity of £a, deferred for n years, on (x), is to be paid for by a uniformly decreasing annual premium, to be extinguished when the annuity is entered upon. Required π, the first year's premium. Here the benefit side is, by (III.), aN(x+n); and, since the premium is to 1 be extinguished in n years, it will make n+1 payments. In order, therefore, that the (n+2)th payment, which would be due when the first payment of the annuity is receivable, may be 0, the annual decrease must be- The payment side π n+1 consequently is, by [20,6], π {N(x − 1) − ‚ + ̧[S(x) − S(x + n + 1)]}. Hence, {N(-1)+1 [S(~) − S(x + n + 1)]}=aN(x+n); n+1[S(x) − Example 8.-An increasing life assurance of £A, £(A+H), £(A+2H), &c., on (a), is to be paid for by a preThe benefit side is, by (XIX.), AM(x) for the first n years, second remainder of life =38·0495 = £38 1s. mium which is to be for the first a years, for the next n years, and for the remainder of life. Required . + HR(x+1); and the payment side is, π[N(x-1) – N(x + n − 1)] ; π[N(x+n-1)-— N(x + 2n − 1)]; }πÑ(x+2n − 1). Therefore, adding these for the whole payment side, we have, π{N(x − 1 ) − } [N(x + n − 1) + N(x + 2n − 1]} = AM (x) + HR(x + 1) ; N(x-1)-[N(x+ n − 1) + N(x + 2n-1)] If A=£100, H= £5, x=30, and n=7, the formula becomes, And, taking the numbers from the table, we should find, This mode of paying premium, although not unusual in practice, is not included in Professor De Morgan's general Problem (I., pp. 10, 11). We subjoin the general expression for the payment side of the equation when a benefit is to be paid for in this manner, that is, by a premium remaining constant ever then the payment side of the equation, for a premium which is to be during the first years, p during the following m years, q during the following n years, and r for the remainder of life, will be, during one or more terms of years, but varying, either by increase or decrease at the end of each term. Let p, q, and r, denote any numbers whatever, either whole or fractional; also, let l, m, and n, denote any terms of years, whatπ[N(x − 1) + (p − 1)N(x + l − 1) + (q − p) N(x+l+ m − 1) + (r− q)N(x + l+m+ n − 1)]. From this expression it appears, that, if p be less than unity, q less than p, and r less than q, the premium will decrease at the end of each term, and vice versa. To apply this expression to the last ex π[N(29) — }N(36) − §N(43) +}N(43)] Example 9.-A life assurance of £A on () is to be paid for by a sum in hand, 6, and an annual premium, π. Required π. Here the benefit side is AM(x), and the payment side is σD(x) + πN(≈ − 1). •*. œD(x)+πN(x − 1) = AM(x). If A= £1000, x=40, and σ = £100, we should find, 18.2308 £18 4s. 7d. Example 10. A person now aged years effected an assurance on his life, of LA, n years ago, at an annual premium of £, of which a payment is just due. He now wishes to dispose of his interest in the same, the purchaser to take on himself the payment of the future premiums, and to receive the sum assured, when it becomes due by the death of (x). Required o, the sum to be paid to (x) for his interest in the policy. The sum to be paid to (*), that is, the present value of the policy, is evidently the difference between the present value of the assurance and that of the future premiums. ..D(x)=AM (x) N(x-1); Whence, σ = A M(x)— N (x−1). D() ample, let x=30, p =}, r=t, l=m=7; and, since there are only two definite periods, during which the payments are uniform, n=0 and q=0. Hence, the expression becomes, =π{N(29) − }[N(36) + ·N(43)]}· Here we must stop. The object of the writer in commencing the present series of papers was, as stated in the outset, to furnish an easy introduction to the papers on the same subject by Professor De Morgan, in the Companion to the Almanack; as he had found, in the course of his experience, that persons even who had paid some attention to the subject, were at a loss to comprehend the scope of the papers alluded to. Whether or not he has in any degree succeeded in his object is for those to say who may have honoured his lucubrations with an attentive perusal. He is himself sensible of many deficiencies in thein, some of which, perhaps, if the opportunity were afforded him, he might be able to amend. Such as they are, however, they are now before the public; and he trusts that, as the work of an amateur, they will be viewed with indulgence. In particular, he has to bespeak the forbearance of the learned individual, in a desire to render whose writings on the subject more accessible than heretofore, they have originated. It is only now, in taking leave of the subject, that he perceives, in its full force, the presumption of which he fears he has been guilty. But as he thinks it may be allowed that his object has been praiseworthy, he trusts this will be received as an atonement for deficiencies in other respects. In the preparation of the articles the writer has made free use of Professor De Morgan's papers, as indeed he could hardly fail to do, since to them and to Mr. Jones's work he owes all the information he possesses on the subject. Notwithstanding this, however, and the circumstance that his papers considerably exceed in length those of the learned Professor, the vein from which the materials have been taken is not nearly ex Sir, From Mr. Zander's reply to my observations on paddle-wheels, any one would suppose I had imagined, that not only the immersed floats, but the whole of the floats in the wheel ought to be considered " effective paddle-board surface" whereas, the whole of my remarks were intended to prove that, not even so much as the total area of the floats immersed, ought to be considered. I feel assured that Mr. Zander has not intentionally so misrepresented me, especially as I observe his letter is of an earlier date than the appearance in the Magazine of my second letter on the subject. My remarks were called forth from facts I have myself observed in numerous experiments made upon paddlewheels; but these were made with wheels having oblique floats, and partially enclosed at the sides, as already described in your Journal. With wheels of this construction, and immersed the depth which I have recommended, I have found that six floats are fully as effective as eight or any greater number, and therefore, conclude that in the common paddlewheel, the total effective area of resistance at any one time cannot exceed the length of one paddle-board multiplied by a, b, (see page 422) the depth of im mersion. Tredgold (The Steam-Engine, Art. 635, 1827 ed.) observes: "The best position for paddles appears to be in a plane passing through the axis; if they be in a plane which does not coincide with the axis, they must either strike more obliquely on the fluid on entering, or lift a considerable quantity on quitting it." This condition is complied with in my improved paddle, by every point of the float radiating in a line from the extremity to the axis. Again, in the same article, he observes: "To set the paddle at any other than a right angle must obviously be a defect; for the resistance to motion becomes less when the surface strikes the water obliquely, whereas, the greater this resistance the greater the effect in impelling the vessel." (otherwise) defect in my wheel is completely obviated by the side plates, which secure a resisting surface of the whole breadth of the wheel at the point of action. This I agree with Mr. Zander, that there is much to be done before we can arrive at a correct theory of the action of steam vessel propellers, especially as to the proper proportion of the propeller to the vessel; upon which very much must certainly depend. I am, Sir, your obedient servant, Wentworth, Dec. 3, 1842. MORTALITY IN THE METROPOLIS, AND IN THE COUNTRY DISTRICTS, ILLUSTRATED. the publication of the "Quarterly Table of the Mortality in 114 of the principal districts of England and Wales," to show the utter falsehood of ascribing an augmented mortality to the metropolis. Sir,-An "augmented mortality in the metropolis" having been recently asserted in your pages, for party purposes, to which I gave an off-hand denial at page 397; I now avail myself of The total number of deaths registered in the metropolis during the quarter ending Sept. 30th, out of a population of In the remaining districts, out of a population of Making the total number of deaths, in a population of . As compared with the average number of deaths in the same quarter of the four previous years, there is an increase of 1,870,727 was 11,019 4,663,808 was 28,050 6,534,535 $9,069 2474 deaths. The average for those years having been 36,595. This, however, includes the metropolis, whose number of deaths was 2 less than the average of the four previous years; so that the whole of the increase has taken place in the country districts. The increase of population should, however, be taken into account, as reducing the proportionate increase of deaths in the country, and also in still farther reducing the diminution in the mortality of the metropolis. The population increases in the town districts about 1.74 annually which would reduce the average of deaths, applicable to the summer quarter of 1842, from 39,069 to 38,208, or 861 deaths less than those actually recorded. But this would still leave an increase upon the quarter of 1,613. In fact, the mortality was 2 per cent. greater than the summer average, which is at the rate of 23 deaths annually in a population of 1,000. In the last summer quarter, the mortality was at the rate of 234 annually in 1,000, viz., 230 in the metropolitan districts, and 23.6 in the provincial towns. The following table shows the principal places in which (in some cases a very remarkable) increase has taken place. RAILWAY ECONOMY-THE SATELLITE LOCOMOTIVE. Now that the miscalculation, blundering, and extravagance, by which the first cost of most of the railways of this coun try has been so enormously enhanced, are evils past, and gone, and remediless; and that the chief thing which share |