fine our remarks to the numerators, and it must be remembered that the denominator in each case is the D corresponding to the present age.

1. When a benefit for the whole life is to be entered upon immediately, the quantities in the numerator have for their signature the present age.

2. When the benefit is not to be entered upon immediately, the quantities in the numerator, (which, as regards the columns made use of, are the same as in the expressions for the corresponding benefits to last the whole life,) have for their signature the age at which the benefit is to be entered upon.

3. When the benefit is to be entered upon immediately, and to last for a term of years only, for example n years, the expressions, as regards the uniform benefits, are derived from those for the same benefits to last the whole life, by writing N(x) −N(x + n) and M (x) – M(x+n), for N (x) and M (x), respectively; and, as regards the increasing benefits, by writing S (x) S (x + n) − n N (x + n) and R (x)-R (x + n) — n M (x+n) for S (x) and R (x) respectively. If we write S (r). S(x+n) and Ŕ (x) (x+n) for S (x) and R (x), we adapt the formulæ to the case in which the increase only is arrested at the end of n years.

1

R

4. The expression for an annuity benefit is changed into that for the corresponding assurance benefit, by writing C, M, and R, for D, N and S, respectively, the signatures remaining the saine, except in the case of C, the signature of which must be diminished by unity.

Since the expressions we have deduced are fractions, the terms of which consist of numbers in the Commutation Table all of the same dimension, none of them rising above the first degree, it follows that, if all the numbers in the table be either multiplied or divided by any one number whatever, the numerical values of the expressions will not be affected.

Hence the truth of the remark in the note, page 442, is established.

And hence also it follows, that the position of the decimal point in the Commutation Tables is perfectly arbitrary. The position assigned to it in our table, is that corresponding to a radix of the Mortality Table of 10,000, although the mortality actually made use of corresponds to a much higher radix, as stated in our first paper, p. 427.

The general problem solved in the preceding pages is, What is the present value of a benefit whose amount is £1? Suppose the problem reversed, and that it is required to find what amount of benefit is equivalent to a present value of £1? The following proportion evidently subsists in all cases: The present value of a benefit whose amount is £1, is to the present value of a like benefit of any other amount, as the amount of the first to the amount of the second. Hence, calling a the present value of the first benefit, and £1 that of the second, we have, a : 1 :: 1 : That is, we

1

α

should be furnished in each case with an expression which is the reciprocal of that which we have obtained for the present value of the same benefit when its amount D (x) is £1. Thus will be the amount, N (x) or annual rent, of a life annuity to be entered upon immediately, and D (x) the sum to be received at the end

M (x) of the year of death, in consideration of a present payment of £1, by an indiviD (x) dual now aged a years. So also, S(x) 2 D (x) 3 D (x) S(x)' S (x)

will be the first, and

&c., the second, third, &c., annual payments of an increasing life annuity whose present value is £1.

Hence the table on page 492, furnishes us also with a solution to the general problem in this form. For, to apply it to any particular case, we have only to form a fraction having for its numerator D (x), and for its denominator the expression in the table corresponding to that case. But this belongs more properly to another branch of the subject.

Before concluding the present paper we make a remark, which perhaps might, with equal propriety, have been introduced elsewhere. It is as to the principle on which the preceding problems have been solved.

Each benefit consists of a payment or payments, to be made at a certain specified time or times, provided a state of things, also specified, shall then have place; such state of things being, where a single life is concerned, either the existence or the non-existence of that life,

coupled in the latter case with the farther condition, that its failure shall have taken place within a certain specified period. Problems I. and X. (using the former or the latter according as the first or the second of the states of things mentioned, is that on which the receipt of the benefit is made to depend) enable us to find the present value of the payment for any or each year of existence; and the sum of the present values for each year that the benefit is to last, is evidently the whole present value of the benefit. Thus, the present value of a life annuity of £1 is the sum of a series, whose terms express respectively the present value of £1 to be received at the end of one year, of two years, of three years, &c., from the present age to the end of life. The present value of an annuity of £1 to last n years, is the sum of the first n terms of the same series; and that of an annuity of the same amount, to be entered upon n years hence, and to last during the remainder of life, is the sum of all the terms after the nth.

The next paper will contain some examples of compound benefits.

Hermes-street, Pentonville.

G.

ever, to encourage the consumption of fuel, by having the back part of the fire as brilliant as the front, unless the heat generated can be made available, and this cannot be done with grates as at present constructed. Three parts of it will be lost, either by being taken up with the passing air, or communicated to the iron and brickwork so far back that it cannot reach the front; so that, at the same expense more heat will be obtained by keeping a large fire, bright in front, but black behind. The remedy must begin with the grate; this might readily be done by having it somewhat like our common register stove, (the frame cast in one piece to prevent escape of gas,) but brought further into the room, and all that part now set in brickwork left The fire exposed to warm the air.

should be lower down-the opening for the smoke at the back instead of the top-the front come down at least 6 inches below the opening, which should not exceed in dimensions 8 inches by 6. The ornamental part may be a matter of taste. A stove grate of this description will not require more than one half the usual consumption of coal with a comfort not shown in the use of the others. I am, sir, Your subscriber,

T. H. B.

FENN'S IMPROVED COACH-WRENCH. Sir,-The coach-wrench was doubtless on its first introduction peculiar to the manufacture from which it derived its name, but now there is scarcely another tool in more extensive employment. Hardly a workshop or manufactory can be visited (however small its complement of tools) in which the coach-wrench, or span. ner, does not play an important part. As hitherto made, however, these implements have been sadly defective, arising partly from the principle upon which they have been constructed, but more particularly from the inferiority of their workmanship, being got up so as to meet the ever constant cry of“ cheap, cheap!"

Some attempted improvements in screw-wrenches have formed the subject of one or more recent patents, but these fall into insignificance before the great improvement achieved in this article by Mr. Fenn, whose novel coach-wrench has

been described at page 391 of the present volume. To this description, I now beg to add a few remarks, in order to set the importance of Mr. Fenn's invention more fully before your readers.

All previous coach-wrenches have consisted of a stem, with one fixed and one moveable cheek or jaw, capable of being set and held at certain distances, according to the size of the bolt or nut to be acted upon; the moving power has ever been a screw, but applied in a great variety of ways. Even when well made, the threads of this screw, after a time, become worn and strip, while in those of the ordinary description this effect very rapidly obtains. Again, with the screw-wrench, when nuts, varying greatly in size, are employed (which is a frequent occurrence) the setting of the wrench to each is a slow and tedious operation, and the continual working of the screw causes most injurious wear of its threads.

In Mr. Fenn's new coach-wrench, no matter how great the discrepancy between the sizes of the nuts, &c., the adjustment is instantaneous, without strain or wear to any part of the instrument. The wedge being free, the jaw A (vide page 391,) is placed against one side of the nut, and the jaw C slips up against the other; on grasping the handle and lever to turn the nut, the jaw C becomes immediately fixed in its position. When the nut is screwed up, the jaw is released by letting go the lever.

The defects and inconveniences of the ordinary coach-wrench have led, in a number of instances, where the work will admit of it, to the employment of spanners of definite fixed sizes, such as half inch, inch, and so forth.

Mr. Fenn's recent improvement, however, has so completely turned the tables in this matter, that his coach-wrench will really be found to be the best, and by far the most convenient that can be employed: inasmuch as it can be applied to nuts of every size within its range (about 4 inches) in less time than one of the particular size required could be selected, supposing it lay before the workman, instead of having, as is often the case, to be sought for.

I am sure your correspondent "T. B.J." will not take it amiss at my adding this testimony to his own, in behalf of the merits and advantages of this simple but really beautiful contrivance of Mr. Fenn's,

duly appreciated.
which only requires to be known to be
I remain, Sir,

Yours respectfully,
WM. BADDELEY.

29, Alfred-street, Islington. November 9, 1842.

MOSELEY'S MECHANICAL PRINCIPLES OF ENGINEERING AND ARCHITECTURE.*

The object of the present volume, by the learned Professor of Philosophy and Astronomy in King's College, is to apply the principles of mechanics to the investigation and solution of the most important and obvious of those questions which present them. selves in the practice of the engineer and architect; and it is by far the most original, ingenious, and useful work which has yet appeared on the subjects which it embraces. The author first treats briefly of those portions of Statics on which the " theory of machines" and "theory of construction" depend, namely, the "parallelogram of pressures "the "equality of moments "-the "polygon of pressures"-the " parallelopipeds of pressures"-" parallel pressures" -and the "centre of gravity." The term pressure Professor Moseley employs in preference to the ordinary one of force; but the difference between them is not very perceptible. Granted, that there is no force till there is pressure; still, the substitution of the one term for the other is but another form of expressing that it is pressure which constitutes force. Indeed, the Professor himself afterwards admits as much, when he says, (Note p. 92,) that " 'pressure and

moving force are but different modes of the operation of the same principle of force."

We are next introduced to Dynamics, or that science which "treats of the laws which govern the motions of material bodies, and of their relation to the forces whence those motions result." Various terms have been used to denote the result of the union of a continued pressure or force with a continued

The Mechanical Principles of Engineering and Architecture. By the Rev. Henry Moseley, M.A., F.R.S, Professor of Natural Philosophy and Astronomy at King's College. With Illustrations on Wood. 8vo., pp. 627. Longman and Co.

motion; such as dynamical effect, efficiency, quantité d'action, puissance mécanique, &c. But, among our French neighbours, all other terms have lately given place to the word travail; and as this has the advantage of great clearness, distinctness, and simplicity, Professor Moseley proposes that we shall follow their example, and make use of the analogous English term, work. The propriety of this recommendation is so obvious, that we make no doubt it will be henceforth universally adopted.

The "work" of overcoming a pressure of one pound through a space of one foot has, in this country, been considered as the dynamical unit, or term in which any other amount of work may be estimated; and in France, the "travail" of overcoming a pressure of one kilogramme through a space of one metre has, on the suggestion of M. Dupin, been called a dyname. Professor Moseley thinks "unit of work" a better term than dyname; and here, again, we think the universal scientific world will go along with him.

For dynamical effect and dynamical unit, therefore, let the terms work and unit of

work be henceforth and for ever substituted; and let the engineering student, who desires to start with clear notions of the elements he has practically to deal with, commit to memory the following definitions.

"Work is the union of a continued pressure with a continued motion. A mechanical agent is thus said to work when a pressure is continually overcome, and a point (to which that pressure is applied) continually moved by it. Neither pressure nor motion alone is sufficient to constitute work; so that a man who merely supports a load upon his shoulders, without moving it, no more works, in the sense in which the term is here used, than does a column which sustains a heavy weight upon its summit."-Page 53.

"The unit of work is the work necessary to overcome a pressure of one pound through a distance of one foot, in a direction opposite to that in which the pressure acts. Thus, for instance, if a pound weight be raised through a vertical height of one foot, one unit of work is done; for a pressure of one pound is overcome through a distance of one foot, in a direction opposite to that in which the pressure acts."-Page 54.

It will be proper further to bear in mind, that by the pound mentioned in these definitions is to be understood the weight of 22.815 cubic inches of distilled water at 62° Fahr., (with barometer at 30 in.,) which is equal to 5,760 grains, or 1 lb. troy, and 7,000 grains, or 1 lb. avoirdupois.

Newton employed the term vis inertiæ to express the inert or passive force which any body opposes to motion; and this term has continued in very general use down to the present day. But as it would be obviously inconsistent with the preceding definitions, to recognize the existence of force (vis) where there is no motion-no pressure -no work done (dynamically considered)— Mr. Moseley abandons the use of that term altogether; and endeavours to make up for it by the following new enunciation of the principle of the vis viva, or moving force.

"The difference between the aggregate work done upon the machine, during any time, by those forces which tend to accelerate the motion, and the aggregate work, during the same time, of those which tend to retard the motion, is equal to the aggregate number of units of work accumulated in the time, if the former aggregate exceed the latmoving parts of the machine during that ter, and lost from them, during that time, if the former aggregate fall short of the latter." -Page 9.

We submit that this is but avoiding one inconsistency, to fall into a greater. The learned Professor has taught us, before, that "a man who merely supports a load upon his shoulders, without moving it, no more works, in the sense in which that term is here used, than does a column which sustains a heavy weight upon its summit;" and yet, he here speaks of the “aggregate work” done by certain "forces," which are of just as passive a character as either the man or the column. It cannot be said that any thing is gained, either in exactness or clearness, by calling the motion obstructive property, which bodies in a state of rest possess, a force which "tends to retard motion," instead of vis inertia. The thing itself remains, though the old definition be shelved.

Neither can it be considered correct to say, that the active forces are those which merely "tend to accelerate motion," for

they cause it as well. And this is consistent with the definition given elsewhere by Mr. Moseley himself (Part I., p. 1) of the term "force," namely, that "force is that which tends to cause or to destroy motion, or which actually causes or destroys it."

The word inertia, (an obvious abbreviation of vis inertia,) has come into such common use amongst us, that it may almost be considered as adopted into our vernacular tongue; and it conveys so intelligibly to every one the thing meant by it, that we think it a waste of labour and ingenuity to go farther a-field in search of a better.

The "work of pressures applied in different directions to a body movable about a fixed axis;" the "moment of inertia; the "acceleration of motion by given moving forces"-the "descent of a body upon a curve"-the "simple pendulum ""—" impulsive force "-" the parallelogram of motion"-"the polygon of motion "_" the principle of d'Alembert," (that in any system of bodies mechanically connected in any way, so that their motions may mutually influence one another, if forces equal to the effective forces were applied in directions opposite to their actual directions, there would be an equilibrium with the impressed forces)-the "motion of translation" -"the motion of rotation about a fixed axis "the centre of percussion "-" the centre of oscillation"-" projectiles "-" centrifugal force”—the “principle of virtual velocities” -"the principle of vis viva," (before adverted to,)—are all successively treated of as branches of dynamics, and with great ability, though in a more scholastic style, (of which more hereafter,) than seems to us absolutely necessary.

We come now to what has been so long a sort of opprobrium mechanicum, the subject of friction, and shall first present to our readers Mr. Moseley's definition of it.

"It is a matter of constant experience, that a certain resistance is opposed to the motion of one body on the surface of another under any pressure, however smooth may be the surfaces of contact, not only at the first commencement, but at every subVOL. XXXVII.

sequent period of the motion; so that, not only is the exertion of a certain force necessary to cause the one body to pass at first from a state of rest, to a state of motion upon the surface of the other, but that a certain force is further requisite to keep up this state of motion. The resistance thus opposed to the motion of one body on the surface of another, when the two are pressed together, is called friction; that which opposes itself to the transition, from a state of continued rest to a state of motion is called the friction of quiescence; that which continually accompanies the state of motion is called the friction of motion."-p. 137.

To this definition, there is this insuperable objection, that it rests on an assumption which in point of fact is not true. All friction is not all resistance to motion; for without friction there would in many cases be no motion. On railways, for example, and in such hydraulic machines as the " Danaide," or Whitelaw and Stirrat's mill, if we were to take away Mr. Moseley's "certain resistance" to motion, there would be no motion at all! Innumerable would be the similar paradoxes in which such a definition would involve us. Something better in the way of definition undoubtedly remains yet to be worked out.

But till all the facts about friction have been incontrovertibly ascertained, it would be idle to hope for any complete definition, and this assuredly has not yet been done. The following summary by Mr. Moseley of the "principal facts" which have resulted from the experiments made on friction, all of them, be it observed, of a very recent date, is excellent in its way; but when the professor adds that "they constitute the laws of friction," he evidently goes further than he is warranted by his premises. He says himself afterwards, (p. 144,) "it is to be regretted that with the means so munificently placed at his disposal by the French Government, M. Morin did not extend his experiments to higher pressures," &c. Why regretted, if "the laws of friction" have been already established?

"The principal experiments on friction have been made by Caulomb, Vince, G. Rennie, N. Wood, and recently (at the expense of the French Government) by Morin. They have reference, first, to the relation of the friction of quiescence to the friction of mo

K K