upon a decreasing life annuity, whose successive payments are to be £10, £9, £8, &c., (that is, a=10, h=1,), it is evident that the 10th payment will be £1, and the 11th 0. And since the annuity is for the whole life, the decrease still goes on, so that the 12th, 13th, 14th, &c. payments will be, - £1, - £2, – £3, &c. That is, the annuitant instead of having anything to receive, will have these sums to pay. It may be also, that the present value of the payments to be thus made by the annuitant, will exceed that of the payments he will have previously received. This is indicated in the application of the formula to any particular case, by its numerical value in that case becoming negative, which will evidently be when h S (+1) is greater than a N (x). A negative value presented by the formula, indicates that the purchase money for the benefit must be paid by the seller. To avoid the inconvenience of the payments becoming negative, h must never be taken larger than n being n any number not less than the number of years during which the annuity is to last. In the case of an annuity for the whole life, the least value of n will be the difference between the age at which the annuity is entered upon, and the oldest age in the Table, when the last payment that can possibly be received will be £h. If in (XIX.) we write x+n for x, we are furnished, as in the case of the simple benefits, with the expression for the (x same benefit to be entered upon n years hence. This expression is, aN (x + n) + h S (x + n + 1).... (XX.) It may also easily be deduced by decomposing the compound benefit. The expression for the same benefit, to last n years, is deduced in the following manner: This modification of the benefit consists of an annuity of La, to be entered upon immediately and to last n years, and of an increasing annuity of £h, £2h, £3h, &c., to be entered upon 1 year hence, and to last n − 1 years, and which is to be either added to or subtracted from the other, according as the benefit whose value is sought is an increasing or a decreasing one. The present value of the first portion is, (IV.), a [N (x) − N (x+n)]; and that of the second portion is found as follows: the present value of an increasing annuity of £h, £2h, £3h, &c., to be entered upon in k years, and to last n years, is, by the Table on p. 492, h[S(x+k) − S (x + k + n) − nN(x+k+n)]. If, therefore, in this expression, we substitute 1 for k, and n-1 for n, we shall adapt it to our present purpose. Making these substitutions, the expression becomes, h[S (x + 1) − S (x + n) — (n−1) N(x+n)]. But (n − 1) N (x + n) = n N(x+n)-N (x+n). Hence the expression becomes, h[S (x + 1) + N (x+n) -S(x+n) - n N (x+n)]. But, by (10.), N (x+n) - S (x + n) = − S (x + n+1). Therefore, the expression becomes finally, h [S(x + 1) − S (x + n + 1) − n N (x+n)]. And if to this we connect by the proper sign the expression for the present value of the first portion of the benefit, we have as the expression sought, + n + 1) − n N (x + n)]. . . . . (XXI.) for n years, and an increasing annuity of £h, £2h, &c., also for n years, and both to be entered upon immediately. The present value of the first is, (IV.), (a–h) [N (x)-N (x+n)]; and of the second, (VIII), h[S(x) − ́S (x + n) − n N (x + n)]. Adding these expressions, we have, (a-h) [N(x)-N (x + n)]+h[S (x) − S (x + n) − n N (x + n)] = a [N (x) − N (x + n)] − h [N (x) − N (x + n)] + h [S (x) − S ( x + n) − n N (x + n)] = a [N(x) - N (x+n)] + h [S (x) − N (x) − S (x + n) + N (x + n) − n N (x + n)]. Now by (10.), S (x)−N (x) = S (x + 1), and S (x + n) + N (x + n) = S (x + n + 1). Hence, the expression becomes, as before, a [N(x) - N (x + n)] + h [S (x + 1) − S (x + n + 1) − n N (x + n)]. It will be seen, on comparing this ex pression with (XIX.), that it does not fol low the same law as the simple benefits in passing from the expression for the value of a benefit to last the whole life to last n years. We shall not seek to deduce here any more of Professor De Morgan's formulæ. We leave the others as a most improving exercise for the student, and pass on to the consideration of a few miscellaneous benefits. LA are to be received by (x) or his representatives at the end of n years, if he be then alive, or at the end of the year in which he dies, if that event take place before the expiry of the n years. Required the present value of the benefit. This benefit is evidently equivalent to an endowment of £A, payable in n years, and a temporary assurance of the same amount, to last n years. Its present value therefore is, by (I.) and (XIII.), A [D (x)+ M (x) - M (x+n)]. (XXII.) This is a benefit of very frequent occurrence in practice. Several of the companies publish tables of the equivalent annual premiums, the method of finding which will be shown in our next paper. Required the present value of a life assurance of £A on (x), with which the sum paid is to be returned. P= r denoting the interest of £1 for a year. The benefit, therefore, is a life assurance of A+ (1+r) P, increasing annually by P; and the expression for the present value will be, by (XIX.), [A + (1 + r) P] M (x) + r P R (x + 1) D(x) Equating this to P, as in the last case, we should find, and d for the remainder of life. Required its present value. If we call the present value of this beAM (x) D()-(1+r) M (x)-r R (x+1)* This and the preceding case illustrate a remark made towards the commencement of the present paper, that when a portion of the benefit depends on the unknown (that is, unknown till the equation is solved) item of payment, the denominator of the expression for the present value is no longer restricted to D (x). These and such like cases, however, belong more properly to that portion of the subject which will be treated in the succeeding paper. The annual rent of an annuity upon (x) is to be a for the first k years, b for the next m years, c for the next n years, Of the second, by (V). third, by (V.) fourth, by (III.) Column S will not serve our turn in this case, since that column is applicable only when the variation in the payments is annual, and equable throughout the whole duration of the annuity. The problem, as proposed, has no such limitations; and we must therefore find the present values of the separate portions of the annuity, and add them together for the whole present value required. The annuity consists of four portions, the present value of the first of which is, by (IV.), a[N(x)-N (x + k)]; c[N (x+k+m) − N (x + k + m + n)]; and of the and the sum of these, [divided, as usual, by D (x),] is the present value required. This sum is, a [N(x) −N(x + k)] + b[N(x + k) − N (x+k+m)] + c[N(x + k + m) — N(x + k +m+n)] +dN(x+k+m+ n) = a N (x) + (b − a) N (x + k) + (c − b) N (x + k + m) + (d − c) N (x + k + m + n) . . (XXV.) This expression shows, as will likewise readily appear from other considerations, that the benefit proposed admits of decomposition into the four following portions, viz., a life annuity of £a, and three deferred annuities of (b-a), £(c-b),and £(d-c,) to be entered upon respectively at intervals of k, m, and n years. Also, that if b, c, and d be respectively greater than a, b, and c, the payments will increase at the expiry of each term, and vice versa. If we adopt the former supposition, and suppose also b-a=c-b=d-c=h, that is, that the annual rents of the deferred annuities are equal, the expression will become, a N (x) + h [N (x + k) + N (x + k + m) + N (x+k+m+n)]; ・ ・ ・ (XXVI.) and if we farther suppose k=m=n=1, that is, that this uniform increase takes place at intervals of one year, it will become, aN (x) + h [N (x + 1) + N (x + 2) + N (x+3)], which, by (6), is equal to a N (x)+h[S(x + 1) − S (x+4)]. And this is the expression we should have derived for the benefit, subject to these restrictions, and for the particular case in which n=4, from the formula [16,12]. Here, for the present, we close our remarks. Hermes-street, Pentonville. G. example, by the perfect straight edges, the plane surfaces, and the cylindric gauges of Whitworth; yet one sort of gauge, that in the most frequent use, remains to this day in all its original rudeness and deficiency. I allude to what is called the wire-gauge, of which, from the mode of their construction, no two specimens are quite alike, even when new, and on which no reliance can be placed, where rigid accuracy is aimed at. Besides this, the arbitrary notation assumed for the sizes, (in some cases the low numbers being affixed to the largest diameters, and in other cases the scale being inverted,) often leads to mistakes. As few publications have a wider circulation, or a more instructive influence on our intelligent mechanics than your Magazine, I venture through your means to call their attention to this circumstance, and to suggest the general adoption of a gauge of such simplicity of construction, that they may be made by any one with great accuracy; and of which the notation of the scale will indicate actual dimensions in decimals, centesimals, or millesimals of an inch, according to the way they may be put together for the peculiar use of the parties requiring them. If such gauges were to be introduced into workshops, they would be found useful in a great variety of cases, besides sizing wire or plate; and as, wherever made, they would all correspond, written orders might be communicated to a distance, with the surety of no mistake arising from discrepancy of measures. I inclose drawings of different forms which may be given to such a gauge; they are all set to indicate centesimals of an inch, but, when wanted for the gauging of very small diameters, the open end of the instrument being set to '05 instead of to 5, it is evident that the same scale, which in the drawing indicates hundredths of an inch, would then represent thousandth parts. A counterpart of this gauge would be useful in cases where the diameters of holes are to be taken; this, of course, may be made of a slip of steel plate, halfan-inch broad at one end, and tapering (by straight line sides) to a point. If the sides, whether long or short, be divided into 50, the scale will then indicate hundredths of an inch, as in the other case; and if this gauge, on being inserted into a hole, penetrate to the division marked 30, a rod which is arrested at the division 30 on the other gauge will exactly fill the hole. I am, Sir, your obedient servant, K. H. MOSELEY'S MECHANICAL PRINCIPLES OF ENGINEERING AND ARCHITECTURE. SECOND NOTICE. The " Theory of the Stability of Structures next occupies Professor Moseley's attention. The equilibrium of any system of bodies-as, for example, a structure of uncemented stones-depends upon two conditions; first, in order that there shall be no turning over, that if a line be drawn through the resisting points of the resultant pressures upon the contiguous surfaces, that line shall be all within the mass of the structure, or, in other words, that it shall actually go through each joint of the structure, avoiding The none; and, second, in order that there shall be no slipping, that the line of direction of the resultant of the pressures upon each of the surfaces in contact shall be within the surface of a right cone, having for its axis the normal (perpendicular) to the common surface of contact, and for its vertical angle twice that whose tangent is the coefficient of the friction of the surfaces. one line is called by Mr. Moseley the line of resistance; and he has every right, which parentage as well as nomenclatural propriety can confer, to give it this name, for he was the first to investigate its properties, (Cambridge Phil. Trans., volume vi.,) and to show that it is, in many important respects, different from the curve of equilibrium of the older writers. The other line is already known to geometrical investigation by the name of the line of pressure. The distance of the line of resistance from the extrados (external face) of a structure, at that point where it most nearly approaches it, Mr. Moseley takes as a measure of the stability of the structure, and calls it the modulus of stability. We agree with him in thinking that this measure or standard of stability is of readier application than the coefficient of stability used by preceding writers. It is a simple rule, and easy to bear in mind, that the more remote the direction of the line of resistance is from the extrados, the less, under all ordinary circumstances of pressure, will the chance of settlement or downfall be. Mr. Moseley very candidly admits, that the idea of this modulus of stability was suggested to him by a rule in fortification, ascribed to Vauban-namely, that, in the construction of revêtement walls, the point where the line of resistances intersects the base of the wall should be distant from its extrados 4ths of the distance between the extrados at the base and a vertical line drawn through the centre of gravity. Professor Moseley proceeds to apply the preceding principles of construction to the various cases of Piers-Walls supported by counterforts and shores-Buttresses-and Walls supporting the thrust of roofs and the weight of floors-all of which are here, for the first time, (we believe,) treated mathe |