FOR in other words, that a machine has no The increased efficiency of the engine power, either of consuming or creating may arise from many other causes than motive power--that it can only transmit that assigned ; and, in the absence of it.” If this is not so, then most certainly better proof than has hitherto been given, "all the experience of the laws of matter, that it does arise from the iso-dynameter, which has been obtained since the use of we may at least be permitted the privilege inductive philosophy, is false," and "our of doubting. whole system of mechanics, since the time The side-thrust at mathematical reaof Galileo, has been resting on a fallacy.” soning comes, as a matter of course, from Let the contrary of this be proved, and all the loss of power ” people. It is but the immortal discoverer of the three laws natural that those whose doctrines reason of motion is proved to have been a teacher condemns should wish to prevent us of sophistry and a propagator of error. from reasoning at all, and leave us at the I am not alone in the strictures I have mercy of every mechanical Sir Oracle, been compelled to make on Mr. Parkes's whose dogmas will not bear the test of too frequent habit of setting up his own examination, unsupported opinion as authority not to I am, Sir, be questioned, often in direct opposition Your obedient servant, to those who ought to be better judges than himself. You, yourself, Mr. Editor, November 21, 1842. have had occasion to notice a case exactly in point, namely, the report on the “ Boccius Light," where (page 299) you MACERONE'S WATER-PROOF BOOTS-OIL justly complain, that “the learned 're ASTRONOMICAL INSTRUMENTS, porters nowhere assign any sufficient reason for the superiority which they are Sir,—The setting in of the “ rainy pleased to assign to the invention they season” will lead the prudent to “ look are puffing. This is exactly what I com to their boots and shoes," and by the plain of, and, by changing a few words, timely application of Col. Macerone's inI might have adopted your very language estimable "water-proof,” preserve to throughout the paragraph. He ought to themselves the superlative comfort of assign a “sufficient reason " why the “ dry feet." substitution of the iso-dynameter causes The several patentees of wood paving a gain of power; but instead of this, ought to unite in presenting the gallant “ how the gain is effected, the reader is Colonel with some slight recompense for left to find out for himself.” the pains he took in paving the way for But your correspondent cannot under their systems. However, neither the stand why I should doubt Mr. Parkes's importance of his advocacy in this matter, inference: I thought I had given my nor his other numerous, if not more imreasons, but I suppose I must have stated portant inventions, seem so likely to imthem unintelligibly; I will repeat them mortalize his name, as his “ water-proof in a more syllogistic form. composition,"'* for the under-standings of Supposing that, from some cause or the human race; a knowledge of the other, the engine does more work now utility of which, I was happy to find, was than formerly : not confined to our own country alone, 1. If this is caused by the substitution but was known and duly appreciated on of the new regulator for the old, it must the Continent. arise from one of three sources, which I While I was recently in Hamburg, I named. heard Colonel Macerone's composition 2. It does not arise from the first or spoken of in the most complimentary second, because no machine can cause manner, and Mr. Campbell, (the agent either gain or loss of power. there for the Mechanics' Magazine,) It does not arise from the third, as ad. informed me that he had adopted a novel mitted by “ A Looker-on." method of application, which had been Therefore, it does not arise from either attended with considerable advantage. of the three ; and, : Instead of brushing the composition over 3. Therefore, the gain is not caused by the substitution of Mr. Lucy's ma • Simply two parts of tallow and one part rosin, chine for the fly-wheel. Q. E. D. the external surface of the boot, he had that he had obtained a most excellent applied it internally. The boot being lubricating material for sextants, and thoroughly warmed before the fire, the other delicate astronomical instruments melted composition was poured in, and in brass, by mixing a small quantity of after turning the boot about, so as to ap rosin with the best olive oil : in the proply the composition to every part of it, portion of one ounce of rosin to a pint of the superfluous quantity was poured out. oil melted together. The oil thus treated The boot was then kept warm until the never turns rancid, nor does it produce composition had been wholly absorbed verdigris when applied to the finest brass by the interior surface of the leather. On work. wearing the boots so treated, the first The oil which is applied to leathern pair of stockings was soiled slightly; hose should be treated with a small the second, not at all; while the boots quantity of rosin ; for if it is sewed this were rendered wholly impervious to wet, will prevent rancidity destroying the carried the most brilliant polish that stitches-if riveted, it will prevent the “Day and Martin" could bestow, and formation of verdigris, which takes place were entirely free from that unpleasant to a considerable extent whenever oil sensation of coldness which is always ex alone is used. Yours respectfully, perienced from boots to which the com WM. BADDELEY, position has been applied externally. 29, Alfred-street, Islington, Mr. Campbell further informed me, November 25, 1842. ON THE CONSTRUCTION AND USE OF COMMUTATION TABLES, FOR CALCULATING THE VALUES OF BENEFITS DEPENDING ON LIFE CONTINGENCIES. Part V.-On the Present Values of Compound Benefits. Compound benefits are those which the problems with which the present and consist of two or more simple benefits; but the following paper will be occupied, to the combinations which may be formed refer to any of the learned gentleman's of these being obviously very numerous, formulæ, which we may not have deduced it would be beside our present purpose for ourselves. Our references will be to attempt giving a complete list of them. made in the following manner, which is Our object will be, in selecting a few of rendered necessary in consequence of his them for illustration, to indicate the me formulæ not forming one consecutive thod of dealing with the more compli series. Formula 10, on page 16, for excated cases, and also to prepare the way ample, will be denoted thus, [16,10]; for the most general application of the formula 7 (2), on page 18, thus, [18,7(2)]; Commutation Tables, which application and so on. will form the subject of the next and As we are no longer to confine ourconcluding paper. A very complete list selves to benefits whose amount is £1, of the formulæ for the more elementary we again point attention to a remark of these benefits, is contained in Pro 11:ade on page 455, to the effect that, fessor de Morgan's first paper on the when we have the present value of a besubject; and as it is hoped that little nefit of £l, that of a like benefit of any difficulty will be experienced with these, other amount will be found by multiply. after the illustrations to which our space ing the first-named present value by the limits us, we shall not scruple, as we number of pounds in the amount in have occasion, in the solution of any of question. As it is convenient to have • It may be of use here to point out a few typographical errors in Professor de Morgan's papers, which might otherwise embarrass the student. First paper, page 11, line 22, for (A+1-1 h), read, (A+-1 H). 16, 29, (a+(n-1) ), at(n-1)h. R(x). 2, (1-v) N (x, y), (1 – v) N (x-1, -1). Also, the terminating braces are omitted in the expressions (18,11) and (19,15 (2)] of the first paper. 25, surance, distinctive symbols to represent the The increasing benefits of which we amounts of benefits of different kinds, have hitherto spoken are those in which we make use of the following for this the annual increase is equal to the first purpose. payment. But the Commutation Tables can also be applied to finding the value s. The amount of an endowment. of increasing benefits, in which the ana. The annual rent of an annuity. nual increase is in no way dependent on h. The annual increment or decrement the first payment; and also of decreasing of a variable annuity. benefits, with the like latitude as to the S. The amount of an endowment as magnitude of the decrease. Thus, a life annuity whose successive payA. The amount of an assurance. ments are to be £a, £(a+h), £(a+2h), H. The annual increment or decrement £(a + 3h), &c., may be decomposed into of a variable assurance. the following annuities, viz., a life an nuity of £a, and an annually increasing Perhaps we may require a few others. annuity, to be entered upon one year If so, we shall explain them as they are hence, of £h, £2h, £3h, &c. The preintroduced. sent value of the first is, (Prob. II.), Also, since the expression for the pre a N (w), and of the second, (Prob. VII.), sent value of a compound benefit is the h S (x + 1). The present value of the comsum of the expressions for the present pound benefit therefore is, a N (-x)+h S values of the simple benefits of which it (x + 1). In like manner it may be is coinposed; and since these are frac shown, that the present value of a life tions having for their denominator D (x), annuity whose successive payments are it will likewise, generally, be a fraction to be £a, £(a - h), £(a - 2h), &c., is, having the same denominator. We shall, a N (x)-h S (x + 1). The following fortherefore, to economise space, usually mula will therefore include both cases, omit this common denominator ; but it the upper sign having reference to the must be carefully remembered, that the increasing, and the lower to the decreasexpressions are incomplete without it. ing benefit. We have said that the expression for the a N (x) + S (x + 1).... (XIX.) present value of a compound benefit is By the remark on p. 493, already regenerally of the form alluded to. The ferred to, the formula for the correspondexception is, (De Morgan, I., pp. 14, 15,) when a part of the benefit depends ing assurance benefits will be, on the unknown item of payment. In A M(x) + H R (x+1).... XIX. this case the expression takes another According to what has been said above, form. When it does so, it will be ex we shall not usually give the formulæ for hibited without abbreviation. the two classes of benefits, since, as we In what follows we shall no longer ad have seen, the formulæ for the one class, here to the formality of problems. For are so readily derived from those for the after reference, however, we shall number other. the expressions we deduce with Roman If in (XIX.) a=h, for the increasing numerals, in continuation of the number benefit, that is, if the annual increase be at which we have arrived in the previous equal to the first payment, the formula problems. becomes, Referring to remark 4, made at the a N (x) + a S(x+1)=a [N (x) + S (x+ close of last paper, (page 493,) we far 1)]=a 3(w), by (10); which agrees with ther premise, that, benefits being divisible (VI), as it ought to do. into the two classes of annuity benefits While the above formula expresses, and assurance benefits, if we deduce the for every value of a and h, the true values expression for a benefit belonging to one of the benefit, yet it must be observed, of these classes, it will obviously be un that in the case of the decreasing benefit, necessary to do so for the corresponding h may be taken so large, that the anbenefit belonging to the other class, since nuitant (we confine our remark to the the relation indicated in the remark quoted annuity, although it is equally applicable always subsists. to the assurance benefit), if he live long We proceed now to the more legiti. enough, will have to pay instead of to mate subject of our present paper. receive. Thus, if a person aged 30 enter &c. upon a decreasing life annuity, whose same benefit to be entered upon n years successive payments are to be £10, £9, hence. This expression is, £8, &c., (that is, a=10, h=1,), it is a N (x + n) + h S(x +n+1).... (XX.) evident that the 10th payment will be It may also easily be deduced by decom£1, and the 11th 0. And since the an posing the compound benefit. nuity is for the whole life, the decrease The expression for the same benefit, still goes on, so that the 12th, 13th, 14th, to last n years, is deduced in the follow&c. payments will be, - £1, – £2, - £3, ing manner : This modification of the That is, the annuitant instead of benefit consists of an annuity of £a, to having anything to receive, will have be entered upon immediately and to last » these sums to pay. years, and of an increasing annuity of £h, It may be also, that the present value £2h, £3h, &c., to be entered upon 1 year of the payments to be thus made by the hence, and to last n-1 years, and which annuitant, will exceed that of the pay is to be either added to or subtracted ments he will have previously received. from the other, according as the benefit This is indicated in the application of the whose value is sought is an increasing or formula to any particular case, by its a decreasing one. The present value of numerical value in that case becoming the first portion is, (IV.), a [N (2)-N negative, which will evidently be when (x + n)]; and that of the second portion h S (x + 1) is greater than a'N (a). A is found as follows: the present value of negative value presented by the formula, an increasing annuity of £h, £2h, £3h, indicates that the purchase money for &c., to be entered upon in k years, and the benefit must be paid by the seller. to last n years, is, by the Table on p. 492, To avoid the inconvenience of the h[S(x + k) - S(@+ k + n) – nN(x+ k + n)]. payments becoming negative, h must If, therefore, in this expression, we subnever be taken larger than, n being adapt it to our present purpose. Making stitute 1 for k, and n-1 for n, we shall these substitutions, the expression beany number not less than the number of comes, h [S (x + 1)-S (x +n) – (n-1) years during which the annuity is to N (oC + n)]. But (n − 1) N (x + n) last. In the case of an annuity for the n N(x+n) - N (x + n). Hence the ex. whole life, the leas: value of n will be pression becomes , h[S (x + 1) +N (1+a) the difference between the age at which - S (x+n) - n N (*+ n)]. But, by (10.), the annuity is entered upon, and the N (x+ S (4 + n)= -S(x+ n +1). oldest age in the Table, when the last Therefore, the expression becomes finally, payment that can possibly be received h [S (x+1)-S (*+n+ 1) - n N (*+ n)]. will be £h. And if to this we connect by the proper If in (XIX.) we write x+n for x, we sign the expression for the present value are furnished, as in the case of the sim of the first portion of the benefit, we have ple benefits, with the expression for the as the expression sought, a [N (X) – N (x+ n)] + h [S (x+1)-S (x+n+1) – n N (ox + n)]..... (XXI.) This expression may also be deduced in for n years, and an increasing annuity of a somewhat different manner,which, as our £h, £2h, &c., also for n years, and both object is illustration, we likewise insert. to be entered upon immediately. The The benefit under consideration evidently present value of the first is, (IV.), (a-h) admits of decomposition into the two fol [N (R)- N (2+ n)]; and of the second, lowing simpler benefits, viz., (confining (VIII.), k[S(x) - S (x+n) - n N(x+n)]. ourselves, for perspicuity, to the increas Adding these expressions, we have, ing benefit,) a uniform annuity of L(a - h) (a -h) [N(x)-N (x+n)] + h [S (x) - S (x + n) – n N (*+n)]= a [N (st) – N(x+ n)] -h [N (2) - N (x + n)] + h [S(x) – S (x+n) -- N (x + n)] = a [N (2) - N (r+n)] + [S (x) - N(x)-S (x + n) + N (x+ n)-N (x+r)). Now by (10.), S (x) - N(x) = S(x+1), and S (x + n) + N (* + n) = S(x+n+1). Hence, the expression becomes, as before, a [N (x) - N (r+ n)] + h [S (x + 1)-S (*+n + 1) - n N (1 + n)]. It will be seen, on comparing this ex low the same law as the simple benefits pression with (XIX.), that it does not fol. in passing from the expression for the value of a benefit to last the whole life nefit P, then £ (A +P) will be the sum to that for the value of the same benefit to be received at death; the benefit, to last n years. Did the law referred to therefore, will be a life assurance of hold here, the signature of N, in the co £(A +P), the present value of which is, efficient of h, would be (x+n+ 1). (See (using the denominator in this case), De Morgan, I., page 21.) (A+P) M (w). And, by condition, we We shall not seek to deduce here any D(a) more of Professor De Morgan's formulæ. We leave the others as a most improving have, P= (A + P) M(x) .. PD(x)= exercise for the student, and pass on to D (2) the consideration of a few miscellaneous AM(*) + PM). By transposition, benefits. £A are to be received by (x) or his P[D (2) - M (x)]=A M (x) Whence, representatives at the end of n years, if he A M (x) (XXIII.) be then alive, or at the end of the year D (2) - M (2) in which he dies, if that event take place If at death, along with the sum asbefore the expiry of the n years. Re- sured, the sum paid is to be returned, quired the present value of the benefit. with simple interest upon it from the This benefit is evidently equivalent to an date of payment, the sum to be received, endowment of £A, payable in ” years, if death take place in the first year, and a temporary assurance of the same will be A+(1+r) P, if in the second, amount, to last n years. Its present value A+(1+2r) P, if in the third, therefore is, by (I.) and (XIII.), A+(1+3r) P, &c., A[D (x) + M (2) - M(x+n)]. (XXII.) r denoting the interest of £1 for a year. This is a benefit of very frequent oc The benefit, therefore, is a life assurance currence in practice. Several of the of A+(1+r) P, increasing annually by companies publish tables of the equivalent P; and the expression for the present annual premiums, the method of finding value will be, by (XIX.), which will be shown in our next paper. [A +(1+r) P] M (x) + P R (2+1) Required the present value of a life D(x) assurance of £A on (x), with which the sum paid is to be returned. Equating this to P, as in the last case, If we call the present value of this be we should find, A M(x) (XXIV.) D(x)-(1+r) M(x) - r R (x+1) This and the preceding case illustrate a and d for the remainder of life. Required remark made towards the commence its present value. ment of the present paper, that when a Column S will not serve our turn in portion of the benefit depends on the un this case, since that column is applicable known (that is, unknown till the equation only when the variation in the payments is solved) item of payment, the denomi is annual, and equable throughout the nator of the expression for the present whole duration of the annuity. The provalue is no longer restricted to D (x). blem, as proposed, has no such limitaThese and such like cases, however, be tions; and we must therefore find the long more properly to that portion of the present values of the separate portions subject which will be treated in the suc of the annuity, and add them together ceeding paper, for the whole present value required. The annual rent of an annuity upon The annuity consists of four portions, (x) is to be a for the first k years, b for the present value of the first of which is, the next m years, c for the next n years, by (IV.), a[N (x) - N(x + k)]; third, by (V.) CIN (x + k + m) - N (x+ k + m + n)]; and of the fourth, by (III.) IN (x+k+m+n); and the sum of these, (divided, as usual, by D (w),] is the present value required. This sum is, a [N(-)-N(x+R)] + b[N(x + k) - N2 + k + m)] + [N(x + k + m) - N(x+k+ m + n)] + N (x + k + m + n)= a N (x) + (6-a) N (x + k) +(c-6) N (*+k+ m) + (d-c) N (x+k+m+n)..(XXV.) + |