tubes being burnt; but, in a fire-box with a flat top, the melting of the lead would only occur when the whole surface was dry, and probably injured. "It is admitted that a locomotive engine should be as light as is consistent with great strength, simple in its construction, composed of as few parts as possible, and that the greatest regard should be had to the diminution of friction; it is thence obvious that four wheels must be preferable to six, provided they carry the engine with the same steadiness. "The use of six wheels originated, (as we have before shown,) in the necessity of supporting the large end heavy fire-box, which was not sufficiently balanced by the smoke-box end; but no such necessity can exist in the locomotives made according to the accompanying plan, as the weight is nearly equally distributed on the front and hind wheels, and not only would two additional wheels be useless, but they would be prejudicial and dangerous when the engines are travelling upon curves. "A four-wheeled engine travelling upon a curve is driven, by the direct application of the moving power, towards the outside of the curve; but, as the wheels are rather conical, the large diameter of the cone will ride on the outside rail, while the smaller diameter of the opposite wheel will bear on the inside rail, and this difference, (as the outside rail is longer than the inside one,) will allow the wheels to revolve without slipping or grinding. With an engine upon six wheels, if the two leading wheels assumed this position, the others would necessarily be dragged after them; but a still more important point is, that the angle which the centre line of the locomotive forms with the tangent of the curve in which it is caused to move is much greater with six wheels than with four, so that the flange of the wheel presses more against the rail with the former than the latter engine. "The pressure against the outside rail, arising from this cause, will be in direct proportion to the distance between the front and hind axle of either engine, so that it will be as 10 to 6. "This pressure and consequent friction is still further increased by the action of the middle wheel, which tends to ride on the same curve as the front and hind wheels, but is prevented from doing so by being in a straight line between the two, and is thus forced to move laterally between the chord and the circumference of the curve. "The friction arising from this lateral motion further presses the engine against the outside rail. Thus the four-wheeled locomotive has, in proportion, a greater weight on the front wheels, it presses less against the outside rail, and offers much less friction when travelling on curves; hence, it has less tendency to be thrown off the rails, it is more simple in its construction, less expensive in repairs on account of this simplicity, and the smaller cost of it fully justifies the directors of the several railways who have given the preference to this description of engine.' At the time the above paper was read before the Society, the four-wheeled engine had but few supporters, arising, no doubt, from the erroneous supposition, that the safety of the engine was in proportion to the number of wheels used. It has, however, been steadily gaining ground in public estination, and from the alterations going on in the construction of the six-wheeled engine, the advocates of them are evidently less confident in their superiority; and it is most gratifying to us that the advantages to be gained by the use of inside framing, which we then pointed out, are now tacitly admitted by our opponents of the greatest practical experience, who are gradually abandoning the outside frame. As the inside frame becomes more and more general, the third pair of wheels will disappear, as not only useless, but really tending very materially to produce those accidents which they are supposed to guard against. Indisputable proof has been furnished, that an engine with inside framing cannot come down by the breakage of an axle: an engine, therefore, is equally safe on that plan of construction whether on four, six, or eight wheels. The advantages of four-wheeled engines, on our plan of construction, we maintain to be the following: 1st. The engine on four wheels is less costly than the one on six wheels; therefore to have the same number of engines, or the same power, on a line of railway, much less outlay of capital is required. 2nd. It allows the engine to be got into less space, consequently it is more compact, firmer, less likely to derangement, and much lighter. 3rd. Though the engine is lighter, the adhesion is more perfect from the weight on the driving wheels remaining nearly uniform, however unequal or out of level the rails may be; but in the engine with six wheels the adhesion is often imperfect (arising from the impossibility of mathematical precision in maintaining rails on the level,) although there may be fully as much weight on the driving wheels generally; that is, the fore and hind wheels sometimes carry the greatest part of the engine. When the driving wheels get into an uneven part of the road, and the constant action of the power of the engine is not resisted by the adhesion at these points, the driving wheels revolve without properly advancing the train, as every observant traveller knows ; and all weight carried beyond what is necessary for adhesion on the rails, is an unprofitable load. There is much less of this in the four-wheeled than the six-wheeled engine, seeing that there is only one pair of wheels used for adhesion both in the four and six-wheeled engine, when used for passenger traffic; but, as the four-wheeled engine is lighter than the six-wheeled engine, there is less power required to take it up the inclines, and therefore more available power left applicable to the traction of the train. 4th. The engine is safer, as it adapts itself better to the rails, not being so likely to run off the line at curves or crossings. 5th. It is more economical in the working, requiring less fuel, there being also a less amount of depreciation, as there are fewer parts in motion, consequently, less friction, or wear and tear, and fewer parts to maintain; and even those are more easily got at, therefore much less expense is incurred in those repairs, which are common to both plans. 6th. The buildings, turn-tables, lathes, drills, smithies, and other costly conveniences necessary for the maintenance and repair of the engines, are not required on so large and extensive a scale, as the engine on four wheels is less in size than the one on six wheels. 7th. As the engine is more simple in its form and parts, there are fewer chances of delays, stoppages, and disappointments during the journeys or the times of taking the trains. Whilst, therefore, those individuals, who have advocated the use of the sixwheeled engines are constantly changing their ideas, at one time adopting a large fire-box with the outside frame, and the addition of a third pair of wheels behind the heavy box to carry it; then changing to the small fire-box, with the third pair of wheels placed before it, and, subsequently, by the tardy adoption of the inside frame; we have been steadily persevering with our original plan, of engine on four wheels, which is now brought to a state of perfection for power and economy far beyond anything we could have expected. In proof of this, we can confidently refer to the London and Birmingham, the Eastern Counties, the Midland Counties, the North Union, the Lancaster and Preston, and the Manchester, Bolton, and Bury Railways, which are worked exclusively with the form of engine we have adhered to; and also to the Edinburgh and Glasgow, Glasgow and Ayr, and Runcorn Gap and St. Helen's, and several other lines which have in part adopted it. In justice to ourselves we have thought it right to lay these remarks before the public, at the same time that we are quite ready to construct engines upon six, or any other number of wheels, freeing ourselves from the responsibility of the consequence of any other plan than our own; and only requesting that such of our friends and the public as may entrust their orders to us will permit us, at least for the safety of travellers, and our own credit, to adhere to inside framing. BURY, CURTIS, AND KENNEDY. Fig. 1, is an elevation of the "Albert Engine;" fig. 2, a plan; A boiler; B fire-box; C Smoke-box, in which are placed the cylinders; DD cylinders, (valve-box, and valves removed;) EE driving-wheels; FF connecting rods; G crank axle; HHHH eccentrics for working valves; II framing of engine; J buffers; K K safety-valves; L, starting and reversing handle; M steam cock for regulating speed; N N brass pumps for supplying boiler with water; 00 pipes for conducting water from tender to pump; PP leather hose between engine and tender; QQ bushes for axles; RR springs; SS guards for clearing rails. ON THE CONSTRUCTION AND USE OF COMMUTATION TABLES, FOR CALCULATING THE VALUES OF BENEFITS DEPENDING ON LIFE CONTINGENCIES. Part III. On the Present Values of the Simple Benefits. We propose in this paper to show how the present value of benefits may be deduced from the Commutation Tables. We must first explain what is meant by the present value of a benefit. As regards the purchaser of a single benefit, it is seldom indeed that he will receive the precise value, (using the word in its ordinary sense) of what he has paid. In the case of certain kinds of benefits indeed, it is impossible that there should in any case be an exact compensation in this sense. For instance, if the benefit be an endowment, that is, a sum of money to be received in the event of the purchaser attaining a certain specified age; if he do not attain that age, neither he nor his representatives will receive anything, and consequently the money he has paid will be lost. On the other hand, if he do attain the specified age, the sum to be then received will be more than his payment would have amounted to, if it had been put out at interest at the time the bargain was made; else he would not have run the risk of losing it in the interval. In the case of such a benefit, therefore, the purchaser must be either a gainer or a loser. If the benefit consist of an annuity, the purchaser may live just such a period as that the number of payments he will receive will be exactly equivalent to the money he has paid. If he live a shorter time he will be a loser, and if a longer he will be a gainer. Similar remarks apply to the case of a life assurance. The purchaser may die, and the assurance be received, at the end of a period, the improvement of his premium or premiums during which would have produced_an amount just equal to the assurance. But as it is much more likely that he will die either before or after the period of this equality, so the probability is proportionally greater that he or his representatives will be either gainers or losers. But while there is this uncertainty in individual cases, there is, in a sufficiently numerous aggregation of cases, a regularity in the occurrences of the events on which the payments of benefits are usually made to depend, which admits of the calculation of the average values of those benefits to a very considerable de gree of nicety. For, if we assume the mortality which will be experienced by a class of purchasers, sufficiently numerous to secure an average, to be the same as that indicated in the table which we take as our guide, (and which table presents the averages of a number of observations)-we say, making this assumption, it is evidently quite practicable to name the sum which will require to be contributed by the whole body of purchasers, to afford to each of them a benefit of a certain amount, on the occurrence of a specified contingency, the time, or the fact of this occurrence, or both, being uncertain, in each individual case. And this sum, divided by the number of purchasers, will be the amount which each ought to contribute, since all at the time of purchase are supposed to be equally likely to come into possession of the benefit. It is the last named amount, obtained in this manner, which is called the present value, or single premium, of the benefit to be purchased. The meaning of the term present value might, perhaps, have been more briefly explained, by defining it as the sum which would be required from each of a large number of purchasers of benefits of the same kind, so that the seller should be in the end neither a gainer nor a loser. We now proceed to the application of the tables. But first we remark, once for all, that in what follows we always suppose the amount of the benefit to be purchased, to be £1; except of course in the cases of an increasing annuity or assurance. In these we suppose that amount only, to which the purchaser becomes or may become first entitled, to be £1. When the present value of a benefit of £1 on any life is found, that of a similar benefit of any other amount, on the same life, will obviously be found by multiplying this present value, by the number of pounds in that amount. Problem I. To find the present value of an endowment of £1; that is, of £1 to be received at the end of n years provided (x)* be then alive. By this symbol is meant, and the phrase may in reading be substituted for it, "a life now aged x years." The number of individuals now alive, of the given age, according to the Mortality Table, is 1(x); and the survivors of these, at the end of n years, is the number represented by the table to attain the age x+n, viz., l (x+n). This number is consequently the number of pounds which will have to be paid at the end of n years. But since money makes interest, the present value of this sum, or_l(x+n)v", contributed now, will be sufficient to provide for the payment of the benefit. And since all now alive contribute equally, the amount to be contributed by each, that is, the required 1(x+n)v"* present value will be 1(x) To adapt this expression to the Commutation Tables, multiply the numerator and denominator by , (which will not alter its value,) and it becomes 1(x + n) v*+N But the numerator of this expression D (21) 3629.2895 D (10) 6051.0605 And if the amount to be received were £100, the present value would be For £100 the present value would be 73427 x 100 = 73·427 = £73 8s. 6d. Problem II.-To find the present value of a life annuity of £1 on (x). This benefit consists of a series of payments of £1, to be made at the end of 1, 2, 3, &c., years, to the end of life. Its D (x + 1) D (x + 2) D (x + 3) D (x) D(x) D(x) present value, therefore, will be the sum of the present values of the several payments. These present values are, by last problem, and so on. And their sum is D (x + 1) + D (x + 2) + D (x + 3) +, &c. D (x) N(x), which is therefore the present value required. • We have assumed above that the number of purchasers of endowments will be the number represented by the Mortality Table to be alive at the age at which the purchase is made. But the only assumption as to their number which it is necessary to make, is that this number will be sufficient to secure an average mortality proportional to that represented in the Table. If we had assumed any other number, we should have had to find by a proportion, the which by (1) is equal to which by (4) is equal to 18.2948 £18 5s. 11d. = + 3, &c., years hence, to the end of life; and its present value will, therefore, be the sum of the present values of these payments. These present values are found by Problem I., and their sum is, + 2) + D (x + n + 3) + &c. D (x) N (x + n), and this is, therefore, the present value required. D (x) blem; which problem is, therefore, but a particular case of the present. And the following rule will apply to both : Divide the number in column N, opposite the age at which the annuity is to be entered upon, by the number in column D opposite the present age: the quotient will be the present value of the annuity. A similar remark will apply to the other benefits, and a corresponding rule for all of them might be formed. But, as our space is limited, the present intimation must suffice. Problem IV.-To find the present value of a temporary annuity of £1 for the next n years, on (x). .... This benefit consists of n payments of £1 to be made at the end of 1, 2, 3, . n, years, if a be then alive. The present values of these payments are, |