use.

Your obedient servant,

THOMAS ROLLS.

123, Albany-road, Old Kent-road, London.

October 20, 1842.

In fig. 3, which is a side view, 1 is the cylinder; 2 2, cranks on shaft through the cylinder; 3 3, main crank; 4, cross head, with connecting-rods to cranks 22; 5, strap head, connected with cross head, 4; 6 6, the connecting-rods of 4 and 22; 7, the eccentric, worked by cog-wheels, 10, in the proportion of 12 to 24, the smaller one being on the main shaft, the larger on the eccentric spindle, so that the shaft makes two revolutions for the eccentric's one; 8, rod from the eccentric, to work the cylindrical slidevalve, 14; 999, stuffing-boxes; 10, the two cog-wheels, in the proportion of 12

to 24; 11, steam-pipe from boiler; 12, waste-pipe; 14, cylindrical slide-valve.

The length of stroke is indicated by the dotted lines ab in fig. 1. As the crank traverses from A to B, the steam crank performs one revolution, and from B to another; and on account of this, which is the distinguishing feature of the engine, I call it the "Double Revolution High-pressure Steam-engine."

T. R.

MR. LUCY'S SUBSTITUTE FOR THE FLYWHEEL.

Sir,-Those who uphold the doctrine of a loss of power by the action of the crank, the fly-wheel, or any other machine, may, with due consistency, consider Mr. Lucy's apparatus as "an improvement of greater consequence than any which has been made in the crankengine for many years." Your correspondent, "A Looker-on," does not, indeed, openly avow himself to be of this opinion; but it appears plainly from his letter, at page 438, that the only way he is inclined to account for the alleged gain of power by the iso-dynameter is, by the old story of the loss from the action of the fly-wheel.

Without meaning any disrespect to your correspondent, I must beg to decline entering into any controversy on this subject. The doctrine, that machines (considered apart from the external circumstances of friction and resistance of the air) can only transmit power, and are utterly incapable of causing either gain or loss in the transmission, has been demonstrated over and over again, in such a variety of ways, both from theoretical and practical considerations, that I am sure, if doubters will not be convinced by what they may find in almost any mechanical treatise, it would be a useless and reprehensible occupation of your pages to reproduce these often-used arguments. I will merely quote the following passage from Mr. Scott Russell's Treatise, already mentioned:-" The great fundamental principle in the construction of machinery is, that the work done depends in quantity only upon the quantity and velocity of the power applied, and not at all upon the form of the machine;

in other words, that a machine has no power, either of consuming or creating motive power that it can only transmit it." If this is not so, then most certainly "all the experience of the laws of matter, which has been obtained since the use of inductive philosophy, is false," and "our whole system of mechanics, since the time of Galileo, has been resting on a fallacy." Let the contrary of this be proved, and the immortal discoverer of the three laws of motion is proved to have been a teacher of sophistry and a propagator of error.

I am not alone in the strictures I have been compelled to make on Mr. Parkes's too frequent habit of setting up his own unsupported opinion as authority not to be questioned, often in direct opposition to those who ought to be better judges than himself. You, yourself, Mr. Editor, have had occasion to notice a case exactly in_point, namely, the report on the "Boccius Light," where (page 299) you justly complain, that "the learned reporters nowhere assign any sufficient reason for the superiority which they are pleased to assign to the invention they are puffing. This is exactly what I complain of, and, by changing a few words, I might have adopted your very language throughout the paragraph. He ought to assign a "sufficient reason why the substitution of the iso-dynameter causes a gain of power; but instead of this, "how the gain is effected, the reader is left to find out for himself."

But your correspondent cannot understand why I should doubt Mr. Parkes's inference: I thought I had given my reasons, but I suppose I must have stated them unintelligibly; I will repeat them in a more syllogistic form.

Supposing that, from some cause or other, the engine does more work now than formerly:

1. If this is caused by the substitution of the new regulator for the old, it must arise from one of three sources, which I named.

2. It does not arise from the first or second, because no machine can cause either gain or loss of power.

It does not arise from the third, as admitted by "A Looker-on."

Therefore, it does not arise from either of the three; and,

3. Therefore, the gain is not caused by the substitution of Mr. Lucy's machine for the fly-wheel. Q. E. D.

The increased efficiency of the engine may arise from many other causes than that assigned; and, in the absence of better proof than has hitherto been given, that it does arise from the iso-dynameter, we may at least be permitted the privilege of doubting.

The side-thrust at mathematical reasoning comes, as a matter of course, from all the "loss of power" people. It is but natural that those whose doctrines reason condemns should wish to prevent us from reasoning at all, and leave us at the mercy of every mechanical Sir Oracle, whose dogmas will not bear the test of examination,

Sir, The setting in of the " rainy season' will lead the prudent to "look to their boots and shoes," and by the timely application of Col. Macerone's inestimable 66 water-proof," preserve to themselves the superlative comfort of "dry feet."

The several patentees of wood paving ought to unite in presenting the gallant Colonel with some slight recompense for the pains he took in paving the way for their systems. However, neither the importance of his advocacy in this matter, nor his other numerous, if not more important inventions, seem so likely to immortalize his name, as his " water-proof composition, * for the under-standings of the human race; a knowledge of the utility of which, I was happy to find, was not confined to our own country alone, but was known and duly appreciated on the Continent.

While I was recently in Hamburg, I heard Colonel Macerone's composition spoken of in the most complimentary manner, and Mr. Campbell, (the agent there for the Mechanics' Magazine,) informed me that he had adopted a novel method of application, which had been attended with considerable advantage. Instead of brushing the composition over

Simply two parts of tallow and one part rosin, melted together and applied warm.

distinctive symbols to represent the amounts of benefits of different kinds, we make use of the following for this purpose.

s. The amount of an endowment. a. The annual rent of an annuity.

h. The annual increment or decrement of a variable annuity.

S. The amount of an endowment as

surance.

A. The amount of an assurance.

H. The annual increment or decrement of a variable assurance.

Perhaps we may require a few others. If so, we shall explain them as they are introduced.

Also, since the expression for the present value of a compound benefit is the sum of the expressions for the present values of the simple benefits of which it is composed; and since these are fractions having for their denominator D (x), it will likewise, generally, be a fraction having the same denominator. We shall, therefore, to economise space, usually omit this common denominator; but it must be carefully remembered, that the expressions are incomplete without it. We have said that the expression for the present value of a compound benefit is generally of the form alluded to. The exception is, (De Morgan, I., pp. 14, 15,) when a part of the benefit depends on the unknown item of payment. In this case the expression takes another form. When it does so, it will be exhibited without abbreviation.

In what follows we shall no longer adhere to the formality of problems. For after reference, however, we shall number the expressions we deduce with Roman numerals, in continuation of the number at which we have arrived in the previous problems.

Referring to remark 4, made at the close of last paper, (page 493,) we farther premise, that, benefits being divisible into the two classes of annuity benefits and assurance benefits, if we deduce the expression for a benefit belonging to one of these classes, it will obviously be unnecessary to do so for the corresponding benefit belonging to the other class, since the relation indicated in the remark quoted always subsists.

We proceed now to the more legitimate subject of our present paper.

The increasing benefits of which we have hitherto spoken are those in which the annual increase is equal to the first payment. But the Commutation Tables can also be applied to finding the value of increasing benefits, in which the annual increase is in no way dependent on the first payment; and also of decreasing benefits, with the like latitude as to the magnitude of the decrease.

Thus, a life annuity whose successive payments are to be £a, £(a+h), £(a+2h), £(a+3h), &c., may be decomposed into the following annuities, viz., a life annuity of £a, and an annually increasing annuity, to be entered upon one year hence, of £h, £2h, £3h, &c. The present value of the first is, (Prob. II.), a N (x), and of the second, (Prob. VII.), hS (x+1). The present value of the compound benefit therefore is, a N (x)+hS (x+1). In like manner it may be shown, that the present value of a life annuity whose successive payments are to be £a, £(a− h), £(a−2h), &c., is, a N (x) − h S (x + 1). The following formula will therefore include both cases, the upper sign having reference to the increasing, and the lower to the decreasing benefit.

a N (x) ±h S(x+1)....

(XIX.)

By the remark on p. 493, already referred to, the formula for the corresponding assurance benefits will be,

AM (x)+HR (x + 1) . . . . XIX. According to what has been said above, we shall not usually give the formulæ for the two classes of benefits, since, as we have seen, the formulæ for the one class, are so readily derived from those for the other.

If in (XIX.) a=h, for the increasing benefit, that is, if the annual increase be equal to the first payment, the formula becomes,

a N (x) + a S(x + 1) = a [N (x) + S (x + 1)]=a S(x), by (10); which agrees with (VI), as it ought to do.

While the above formula expresses, for every value of a and h, the true values of the benefit, yet it must be observed, that in the case of the decreasing benefit, h may be taken so large, that the annuitant (we confine our remark to the annuity, although it is equally applicable to the assurance benefit), if he live long enough, will have to pay instead of to receive. Thus, if a person aged 30 enter