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The first column contains the age at which the assurance is effected; the second the annual premiums given by I. M., and the other three the premiums according to the mortality tables and rate of interest indicated, without any addition in the shape of commission.
On comparing 1. M.'s premiums with those in the other columns, it will be seen that they correspond most nearly with those calculated by the Northampton Table; that is, the differences between the corresponding premiums are in that case the most regular, although there are, even here, discrepancies enough to show, either that the Northampton Table
cannot have formed the basis of the calculation, or that a different rate of commission has been added to the premiums at different ages.
However, the corres. pondence is near enough to enable us to form a judgment as to the sufficiency of the rates.
I. M.'s premiums (I call them so for brevity) then, it will be seen, with an average addition of about 6 per cent., correspond pretty nearly with those deduced from the Northampton Table. Now, supposing the Northampton Table to give a tolerably correct approximation to the mortality which will be experienced amongst the assured at the ages under
consideration, and that 3 per cent. is the depends as well on the progressive rate highest rate that can be calculated upon of mortality between the same ages. It for the safe investment of money (which will be safer, therefore, to calculate the in the present state of the market is the premiums for such assurances by tables fact)—I say, making these suppositions, which indicate a lower rate of mortality it will be allowed by all who have paid than the Northampton Table. any attention to the subject, that an ad Such are the Carlisle and the Governdition of 6 per cent to premiums calcu ment Tables, the premiums derived from lated on these data is wholly inadequate which I have given above. It will be to provide for expenses of management seen that these premiums differ but little and risk of Auctuation in the rates of both from each other; while they are, in every mortality and interest, to say no hing of case, higher than the Northampton preprofits to members or share-holders. On miums, and generally a mere fraction these grounds, therefore, I think we are under those given by Iver M‘Iver. warranted in saying that the premiums Hence, the conclusion becomes still more are inadequate.
unavoidable, that the last-named are inBut it is well known that the Nor adequate. But are the tables from which thampton Table indicates a rate of mor these higher premiums have been detality, especially at the younger ages, duced to be depended on, as giving a (which are just those with which we correct representation of the mortality have at present to do,) far greater than which will be experienced at the ages has ever been experienced in any assur under consideration ? As regards the ance society. By the showing of Mr. Carlisle Table, Mr, Milne, by whom it Morgan himself, who was the great pa was compiled, states that, in consequence tron of this table, the number of deaths of the introduction of vaccination, since between the ages of 10 and 20, in the the observations on which the table is Equitable Society, during an experience founded were made, the mortality indiof fifty years, were only half those pre cated in the table at the early ages is dicted by the Table referred to. And greater than may generally be expected. certainly when, as in the case before us, And the Government Table we may preit is the interest of the parties effecting sume to be in a similar predicament. the assurances that the lives put in shall Hence, it will be unsafe to make use of be as good as possible, it were folly to the premiums derived from these tables, calculate on a greater rate of mortality without a considerable augmentation; than other offices have experienced in less and hence, also, an office which should favourable circumstances. But the effect confine its business to assurances of the of calculating in assurances of the kind kind under consideration, at the rates under consideration, according to a higher given by I. M., could hardly fail to be rate of mortality than will actually be ruined. experienced, is, to give the annual pre It may be satisfactory to some of your miums under their true value, since more readers that I should explain the method will be alive to claim their endowments by which the foregoing table has been than the table predicted, and conse calculated. The calculation was made by quently, than the premiums were intend means of the formula which I gave in a ed to provide for. It is true that, when previous communication, at page 117 of the benefit is to be paid for by annual your present volume; and the elements, in premiums, a compensation to some extent the cases of the Carlisle and Northampwill take place ; since, if the assured die ton Tables, were taken from Mr. Jones's off more slowly than was anticipated, work on Annuities; and in the case of more premiums will be received ; and it the Government Table, from a table conis even possible that in particular in structed from the data contained in Mr. stances the compensation will be exact. Finlaison's Report on the Mortality of But an exact corapensation at all ages the Government Annuitants. (Parliacan never possibly take place: since, mentary Paper, No. 122, 1829.) The leaving the rate of interest out of view, formula referred to is, the amount of the benefit depends solely
8 D (v + n) on the aggregate mortality between the age at which the assurance is effected,
N (x - 1) - N (x + n - 1) and that at which the benefit becomes which in this case becomes, s being equal due, while the amount of the premium to 100, and a + n = 21,
100 D (21)
formerly referred. These articles, howp (x) = N (x – 1) - N (20)
ever, avowedly contain almost nothing of
demonstration; and it is of this deficiency The values of this expression for each that I chiefly complain, and which I value of x, from x = I to x = 10, are should make it my study to supply. the premiums required.
May I, notwithstanding the length to In using the Commutation Tables, not which the present article has insensibly only is the mode of operation more grown, still beg space for a remark or simple, generally, than by any other two on the manner in which Mr. Scott method; but, when we have a series of has chosen to solve Iver M‘Iver's for. values of any given benefit to calculate, mer problem? Mr. Scott seems to enwe almost always meet with facilities, tertain a most magnanimous contempt which, besides materially abbreviating for the labours of his predecessors in this our labour, also afford the means of branch of science. The results of these proving the accuracy of part, at least, of labours have been published expressly our work. Thus, in the case before us, for the purpose of facilitating the enthe numerator of the foregoing expres quiries of those who should come after ; sion is constant for all the values of x, yet Mr. Scott stubbornly refuses to avail and the denominator alone varies. And himself of them, and gives clumsy theowe observe that the difference between rems for the solution of particular cases, any two successive values of the denomi. which
be solved in their utmost genator, (those corresponding to x = l'and nerality at one-tenth of the labour, by x = 2, for instance,) is N (0) - N (1.) availing ourselves of the tables which Now, by the construction of the Corn have been published. In finding the mutation Tables, N (0) - N(1) =D (1.) value of a temporary annuity for nine Consequently, if the value of the denomi years, which was necessary for the solu. nator corresponding to x=1 be found, that tion of the problem referred to, Mr. S. corresponding to x 2 will be found by chose to do so (page 114) by a method subtracting D (1) from the first value; which involved nine divisions by as many also that corresponding to x = 3, by sub different divisors of seven figures each, tracting D (2) from the second value; and one addition of nine lines; while the and so on for the rest. And the verifi value might be found from the published cation is afforded by comparing the value tables, as I showed in my first solution, corresponding to x = 10, obtained in this (page 116,) by two multiplications, one manner, with the same value obtained by division, and a subtraction. By my sesubtracting N (20) from N (9). Those cond solution, only one subtraction and who are accustomed to calculations of one division are required for the whole this kind will appreciate the facility af solution. Moreover, had the duration of forded by the mode of operation I have the annuity been doubled, so also would described, as well as the advantage of the Mr. Scott's labour, while that by the final verification. When the denomi. method I employed would have undernators are thus found, their logarithms, gone no increase. If the facilities af. subtracted successively from the loga forded in calculation by the use of logarithm of the numerator, give the loga rithms be not too great to suit Mr. rithms of the required values of p. Scott's taste, he will of course, when he
So obvious are the advantages of the has occasion for them, if he act up to his employment of the Commutation Tables, principles, disdain to take them from a for the calculation of the values of bene- published collection, but will insist on fits depending on life contingencies, that calculating them in every case for himself! ! I believe their being so little known and Seriously, Mr. Scott ought not to act used for this purpose arises from the in this manner. If every one were to do want of a systematic treatise upon the so, and refuse to avail himself of the la. subject. To contribute in some degree bours of his predecessors, in matters of towards supplying this deficiency, I am mere drudgery, the advances of science tempted to offer you a few short papers would be slow indeed. I trust I have in illustration of their construction and said enough to make this obvious. applications. In drawing them up, I I am, Mr. Editor, respectfully yours, should be largely indebted to Professor de Morgan's able articles in the “Com
G. panion to the Almanack,” to which I
September 21, 1842.
MR. HALL'S COAL-CONSUMINI CLAIMS.
Sir,--In forwarding you my first letter in reply to Mr. Samuel Hall's advertisement, as it appeared in the Mining Journal of the 10th instant, and which you will be pleased to insert also as an advertisemeni, * I have to request your readers will suspend their opinions with respect to the several points not yet commented on by me, assuring them that none shall remain unnoticed. They will perceive that the use of coal in locomotive engines, on Mr. Hall's plan, is, in the opinion of almost all, (Mr. Hall himself, of course, excepted,) attended with such uncertainty and expense as to be impracticable. I hope hereafter to give the reason why. I may here just observe, that, although on many railroads coal is used with advantage, except as regards the nuisance from smoke, yet, the moment we attempt to burn the 10,000 or 12,000 cubic feet of gas which each ton of coal evolves, the admission of the enormous volume of air which such gas demands, induces so many difficult chemical conditions, and such a complex process, as almost to defy our efforts in effecting complete combustion, and avoiding the nuisance of smoke. The subject is one of great difficulty, in a chemical point of view; and I propose, hereafter, directing attention to its details, as regards locomotive engines. In land boilers, and the greater number of marine boilers, the difficulty has already been surmounted. I am, Sir, yours, &c.,
C. W. WILLIAMS. Liverpool, September 21, 1842.
IMPURE WATER THE CAUSE OF FOUL CIS
TERNS — IMPROVEMENT IN CONSTRUCTION OF CISTERNS SUGGESTED.
Sir,-Controversy on practical subjects generally evolves something of public utility. One thought suggests another; and your Periodical, by giving expression to diversi., ties of opinions, leads on discussion, until the public obtain some beneficial result.
Your correspondents, B. and Mr. Badde. ley, seem to think differently about the chief sources of the impurities of the water in our cisterns, and, consequently, about the remedy to be applied. It is an occurrence so frequent as to have become an axiom, that when a man finds he has the worst of an argument, his sensitiveness is more remarkable than his sense; and the last communication of Mr. Baddeley is, I think, an instance of this kind.
Mr. Baddeley would have us believe, 1. That the water comes into our cisterns comparatively pure; 2. That it gains its chief impurities from the deposits retained in the cisterns; 3. That these deposits are not from the water supplied, but from the spiders, moths, flies, blacks, and such-like nuisances as abound in the places where cisterns are generally fixed; 4. That if we were careful to keep our cisterns clean, we should have
THE SUBSTITUTE FOR A FLY-WHEEL, AT
MR. LUCY's, BIRMINGHAM. Sir,-Would any of your readers be so kind as to describe the substitute for a flywheel, weighing 24 tons, removed from Mr. Lacy's steam-engine, of Birmingham, and by which the performance was increased 10 per cent., as stated by Mr. Parkes in the course of the discussion on Mr. Mosley's Indicator, at the Institution of Civil Engineers, as reported in your Magazine, at page 139 of the present volume. The discussion in your papers relative to the loss of power in the crank might be benefited by the information. Yours, &c.,
A LOOKER-ON. See Cover of this Number; also, Cover of the Monthly Part for September.-Ed. M. M.
In answer to the first of these propositions :-There is the fact, that the water as it comes, not from the cistern, but from the main, is so thick as to be unfit to drink, or to use in the preparation of food ; and that this water, by being undisturbed for some time, becomes clear-that is, through the impurities, which occasioned its muddiness, settling down to the bottom of the vessel.
This fact militates as much against the remaining three propositions as it does against the first. If we get a pail of water from the main, and let it settle, the bottom of the pail receives the deposits; if the same water comes into our cisterns, they receive the deposits. These deposits are increased by every supply of water. Most people would reckon that three days of such deposits as these would be equal to a hundred days of such accidental deposits as Mr. Baddeley speaks of; and it is because they do reckon in this way, that they deem it labour in vain to clean their cisterns out, in expectation of getting clear water. Give them clear water into their cisterns, and nine in ten would take pains to keep them clean. But now, if they were to clean out their cisterns every day, they would be still as far as ever from obtaining pure water.
These reflections on the deposits in our water cisterns have suggested to my mind the possibility of devising another remedy
besides that of Mr. Stuckey. Under the best system of filtration, deposits will take place; and it would be a great improvement if, by some change in the shape of the bottom of the cistern, or in the shape and position of the cock, these deposits could be prevented from accumulating. At present, there is between the cock and the bottom of the cistern a perpetual substratum of stagnant water, which forms a breeding-ground for worms and other animalculæ.
All your readers know that Mr. Baddeley possesses much ingenuity, and perhaps, in his zeal for the public good, he may supply some remedy for this evil ; so that, by the time the public obtained Stuckeyfied water, they might also have butts which would not retain deposits so badly. I am, Sir, yours respectfully,
J. COLE. Old Kent-road, September 27, 1842.
WATER QUESTION-STUCKEY'S FILTRATION v. BADDELEY'S CISTERNAGE,
Sir,-I am glad to perceive my “veteran" opponent, (Mr. Baddeley,) is anxious to have it believed that he is neither influenced by a desire to serve the Water Companies, nor to deprecate the forthcoming Parliamentary Inquiry into their “sins of omission." I at once apologize for having so far libelled him as to imagine that he had such objects in view ; but perhaps he will pardon me while reflecting that he himself acknowledges that he has been "driven into the position of an apologist for the Water Companies," however " uncongenial” such a task may be to his feelings, and the more especially, if I happen to demonstrate that the case he attempts to make out, of the comparative innocence of those Water Companies in producing the filthy and foul beverage their customers swallow, is so utterly against both science and evidence, that no man who knows any thing of Mr. Baddeley's talent and information would for a moment believe that he could honestly hazard such assertions, or profess such opinions. Nevertheless, I was wrong-1 confess it. Mr. Baddeley, in making the statements he has done, that “nine-tenths” of the bad water arise from the laches or neglect of the customers, and not one-tenth ” from any fault of the Companies—that, in truth, in defying me “to produce a single case of the slightest indisposition produced by water as supplied by any of the public Companies," he was really, all the time, a decided water reformer-a“ veteran" associate-one who wisbed to induce or compel the Water Companies to adopt Mr. Stuckey's plan of filtration ; nay, to give him “a push behind on any
stop-short movement." It may be so-it must be so--and the only way to account for it is, on the hypothesis of Butler, that Mr. Baddeley belongs to that species of cha. racter described as men
who may have wit, But who are shy of showing il!" Like the prophet of the heathen king, he blesses those whom he intended to curse : meaning to fight on any side, he commences operations by giving up the citadel. He thinks the Water Companies ought to amend their water;" but then, “there is not a single case of the slightest indisposition produced by it!"
Call you this a " backing of your friends," Mr. Baddeley ? Truly, they are very badly backed !
Why should the Water Companies amend their water, if no indisposition be produced by it?
Well, however, friend or foe, Mr. Badde. ley's defence is a bold one, and it shall be boldly met. He shall have evidence so strong, so full and so complete of the fact that he defies me to prove, that, if he be not convinced, "the eye of his faith” must be " dim" indeed ; so much so as to make an unkind public suspect that he has shut it, on purpose not to see.
A noble lord who has gained some laurels in America just now, before he went to smoke a pipe with brother Jonathan, said some 18 months ago that the water supplied to him was so bad and filthy that saying nothing about drinking it, he positively would not take a bath in it, no matter which of the Companies supplied him. I make no war on individual Companies—such, however, was the fact. The head of the great house of Baring thought differently on this subject from the head of the great house of Baddeley.
But now for proofs, as to the two points of Mr. Baddeley's letter,- 1st, that the water supplied by the Water Companies is deleterious and destructive to health, and 2nd, that when water is delivered pure into cisterns it remains so for a great length of time. I propose to adduce extracts from the evidence as given before Committees of both Houses of Parliament, and before the Royal Com. missioners of the following among other individuals, Dr. James Johnson, Dr. Ker. rison, Dr. Bostock, Dr. Paris, Dr. Yeates, Dr. Wm. Somerville, Dr. Hoffman, Dr. Lind, Dr. Mead, Dr. Wm. Lambe, Aber. nethy, and a host of non-medical men prove the deleterious, dangerous, and fatal effects of that very water which Mr. Baddeley, in his innocence, declares never to have produced the slightest indisposition;" and in the second place to establish my position,