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called "fur" on the bottom and sides of the tea-kettle, I partly believe it is so. So, there goes the carbonic acid gas, which Mr. Cole wishes to preserve, and there goes the lime, which my naval friend has just popped in; and I am thinking that if the lime, in its descent, should just carry down all the monsters and creeping things, and what not-then, in that case, we shall have the very ocular realization and antitype of the muchdubitated chalk formation before our eyes--constructed by the genuine Neptuno-Plutonic process-abounding in those curled and wriggling monsters which populate and characterise it-producing that pure, and purling, and delicious water, which flows every where from the foot of chalk rocks and the spouts of tea-kettles; and proving, most incontestably, the truth and true use of all the systems together of the geologists and the water-filterers.

I have the honour to be, Sir,

Your humble servant,

November 22, 1842.

S. R. B.

ACTION OF WATER ON LEAD. [Abstract of a paper by Professor Christison, read before the Royal Society of Edinburgh.] The author showed the deleterious action of very pure water on lead, and that the purer the water, and the more free of salts in solution, the more powerful was its action on that metal. He mentioned one instance, in which the water was conveyed in a lead pipe from a distance of about three-quarters of a mile, from a spring of extraordinary purity, its total saline ingredients being only a 22,000th part. Here the water acted so powerfully on the lead, that in a short time the cistern in which the water was received was covered with loose carbonate of lead, and the metal could easily be detected in the state of oxide dissolved in the water.

In another instance, where the water was conveyed about half a mile, the same phenomena occurred; but with the additional circumstance, that, in consequence of the impregnation not having been detected in time, as in the previous case, the disease called Colica Pictonum broke out in the house supplied with the water. In this case the water contained no less than a 4500th part of saline matter, but chiefly muriates, which the author had previously found not to pre

vent the deleterious action unless present in much larger quantity.

The author then explained in what manner the action of the water on the lead was put an end to in both instances. In the first case the water was allowed to remain stationary in the pipe for four months, till a firm crust of mixed carbonate and sulphate of lead cystallized on the interior of the pipe, after which no farther action took place. In the second case the pipe was kept filled with a solution of phosphate of soda, consisting of a 27,000th of the salt.

He then stated the following practical conclusions to be drawn from his inquiries, as to the use of lead in conveying water:

1. Lead-pipes ought not to be used for the purpose of conveying water, at least where the distance is considerable, without a careful chemical examination of the water. 2. The risk of a dangerous impregnation of lead is greatest in the instance of the purest water.

3. Water which tarnishes polished lead, when left at rest upon it in a glass vessel for a few hours, cannot be safely transmitted through lead pipes without certain precautions.

4. Water which contains less than about an 8000th of salts in solution cannot be safely conducted in lead pipes without certain precautions.

5. Even this proportion will prove insufficient to prevent corrosion, unless a considerable part of the saline matter consists of carbonates and sulphates, especially the former.

6. So large a portion as a 4000th, probably even a considerably larger, proportion, will be insufficient, if the salts in solution be in a great measure muriates.

7. In all cases, even though the composi tion of the water seem to bring it within the conditions of safety now stated, an attentive examination should be made of the water, after it has been running for a few days through the pipes; for it is not improbable that other circumstances, besides those hitherto ascertained, may regulate the preventive influence of the neutral salts.

8. When the water is judged to be of a kind which is likely to attack lead pipes, or when it actually flows through them impregnated with lead, a remedy may be found, either in leaving the pipes full of the water, and at rest, for three or four months, or by substituting for the water a weak solution of phosphate of soda, in the proportion of about a 25,000th part.

ON THE CONSTRUCTION AND USE OF COMMUTATION TABLES, FOR CALCULATING

THE VALUES OF BENEFITS DEPENDING ON LIFE CONTINGENCIES.

Part IV. On the Present Values of the Simple Benefits-continued.

In the present paper we are to deduce the expressions for the present values of the assurance benefits.

Problem X.-To find the present value of an endowment assurance of £1 on (x); that is, of £1 to be received n years hence, provided (x) shall have died in the preceding year, viz., in the nth year from the present time.

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Of the 7 (x) individuals, whose present age is x years, represented by the mortality table to be now alive, (x + n − 1) survive n 1 years, and (x + n) survive n years. Consequently, 7 (x + n−1) (x+n) is the number who die in their nth year; and it is also the number of pounds which will have to be paid, at the end of n years, to the representatives of those who thus die. The present value of this sum, therefore, that is, the sum which, put out at interest now, would in n years just amount to the firstnamed sum, is what must be advanced now to provide for this payment. This present value is [(x + n − 1 ) − 1 ( x + n)] v". And since all the 7(x) individuals now alive are equally interested, all of them contribute equally to this amount. contribution of each will be, therefore, [l (x + n 1) − l ( +n)] *

1(x)

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The

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rator and denominator of this expression by va, and it becomes

[1 (x + n · 1) − 1 (x + n)] v*+n 1(x) v

But the numerator of this expression is equal to C (x + n 1), and the denominator to D (x), by (1) and (3). Hence, the expression for the required present value, by the Commutation Table,

is

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Since C is not exhibited, we may use for it either of the expressions (10) or (12). Taking the first, the formula becomes,

540-81079 522.65022

2257-6521

=

18.16057 2257-6521

If the sum to be received be £100, its present value will be 008244 x 100 = 8244 = 16s. 6d.

We have seen that when n = 1, that is, when the sum assured is to be received a year hence, provided (x) be then dead, the formula for the present value becomes

C (x)
D (x)

But, by (12), C (x) = v D (x) – D (x + 1).
Hence, by substitution, the formula in
this case becomes
v D (x) − D (x + 1)

=

v

D (x + 1)
D(x)
D (x)
And this expression affords a proof of

⚫008244 2d.

the correctness of the formula in this
problem. For, since v is the present
value of £1, to be certainly received a
D (x + 1)
year hence, and
is (by Pro-
D (x)
blem I.) the present value of £1, to be
received at the same time, provided (x)
be then alive, the difference between
these two is evidently the present value
of £1 to be received a year hence, if (x)
be then dead.

Problem XI.-To find the present value of a life assurance of £1 on (x);

that is, of £1 to be received at the end of the year in which (a) dies, whenever that event may happen.

We saw, by the last problem, that C (x) is the sum which must be contriD (x)

buted now by each of the (x) individuals, to provide for the payment of £1 to the representatives of each of those of their number who die in the first year; C (x + 1) is the sum each must and that

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D (x) contribute to provide for a like payment to the representatives of each of those who die in the second year. And in like manner by making n successively equal to 3, 4, &c., to the end of life, in the formula deduced in last problem, we should obtain, for the amounts to be contributed now, to provide for the deaths that take M (20) 1131.76185 D (20) 3818-9594

Ans.

=

Example.-Required the present value of a life assurance of £1 on (20).

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If the amount assured be £100, its present value will be,

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Problem XIV. To find the present value of a deferred temporary assurance of £1 on (x); that is, of £1 to be received at the end of the year in which (x) dies, provided that event take place in then years following the next k years.

The present value required in this problem, is evidently the sum of n terms of the general series of problem XI. beginning with the (x + k)th; which sum is, by (9), equal to

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Here x =

30, k

M (40)

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D (30)

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2257.6521

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Problem XV. To find the present value of an increasing life assurance on (a), which is to be £1 if death take place in the first year, £2 if in the second, £3 if in the third, and so on; the assurance increasing each year that the payment is deferred, by the amount of the first year's assurance, till the end of life.

The present value here is found exactly in the same way as that of the corresponding annuity benefit in Problem VI. For this assurance may be conceived to be made up of a series of assurances of £1, of which the first is to be entered upon immediately, and the others

M (x) + M (x + 1) + M (≈ +

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9.7582

=

£9 15s. 2d.

at intervals of one year, until the life shall have ceased to exist. The sum of the present values of all these assurances will evidently be the present value required. Now the present value of M (x)

the first, is, by Problem XI.

D (x); and of the second, third, fourth, &c., by Problem XII.,

M(x+1) M(x+2) M (x+3), &c.; D (x)

D(x) D (x)

D (x)

and the sum of all these, that is, the present value required in the problem, is, by (2),

2) + M (x + 3) + &c.

R (x) D(x)*

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£3 if death take place in the third year, 3 C (x + 2) and so on. And the sum

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D (x) of these is, by (11),

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Example. Required the present value of an increasing assurance of £1, £2, £3, &c., on (80).

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fore unnecessary to repeat, the expression for the present value of the benefit in

the present problem is found to be

R (x + n)
D (x)

Example. Required the present value of an increasing assurance of £1, £2, £3, &c., deferred 15 years, on (30.)

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