These values will be required in aid of the investigation, and are therefore put down together.

(a—b) (a'-b')
a'b-a b'

ME=EK sin (w+ x)=KP sec x sin (w+x),

The points E, F give the same result, as do likewise E', F'; but these two results are constituted of different elements.

d{(a−b) cos — (a−b') cos o'}sin w=2(a−b) (a′ — b′) cos o cos 'cos w...(10)

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duction, so as to eliminate from the result; for (10) is readily changed into

and equating this with the result in formula (3) we have a neat expression of the required condition, viz.,

c{ (a'—b') sec p'—(a−b) sec ø }=2(a'b—ab').............. (12)

Scholium.-This condition is remarkable from its independence of the length,

PROBLEM X.

From the same data as before, to find the volume of the cutting, contained between the boundaries PP and QQ'.

d, of the cutting; although a little consideration would really lead us to expect it.

This solid APQBA'P'Q'B' is, obviously, equivalent to the following combination of the four tetrahedra :

APKG-BQKH-A'P'K'G' + B'Q'K'H'.. .. .. .. .. (13)

Now, by the ordinary principles of mensuration, we have the following values:
APKGAE. KP. PG= a sin . KP2 tan w,

As I have no intention of computing special tables for facilitating these calculations of cuttings, I may be excused from entering into the decompositions or modifications of this formula. Several such are, however, obvious enough to those who may wish to use it, and especially to those who are in the habit of computing tables. The tables of Messrs. Bidder and Creedy, which you reviewed a short time ago, (and of the formulæ for which you gave an elegant investigation,) will suggest more than one.

Other

tables, too, have been published by Sir John Macneil, Mr. Hughes, and Mr.

a-b

•(aï3—¿′3) sin p'}, or

a-b'

sin p − (a−b) (a'2 + a'b'+ b2) sin p′ }.... (15)

Fig. 7.

and hence that case must be treated on other principles. No method has occurred to me more simple and elegant than that which you have yourself given.

When -0°, or the azimuths of the surface-plane and railway differ by 90°, the expression again takes the illusory form; in this case, therefore, a separate investigation becomes necessary, and is annexed. To all other cases the formula (15) is immediately applicable.

As the two profile sections are in this case equal in every respect, the cutting becomes a prism, whose end is the figure APP'A'.

Now, producing the lines AP, A'P' to meet in C, we have

୯

The required volume is the product of this by the length, d, of the road under consideration: viz.

2 { c (sin o + a sin p') + a a' sin (ø+4') } cosec (+).... (16)

When any special relations exist amongst the data, they generally tend to simplify the formulæ and lessen the labour of construction. These, however, as they will readily occur to every one who wishes to use the method, need not be pointed out here; and it will be seen, that I have always taken the most gene

Fig. 1.

ral case of every problem, that was compatible with the surface of the ground within the limits of the cutting, being a plane. It may form the subject of another paper to discuss the problem when this condition is not fulfilled.

WHITE'S RAILWAY LINK.

(To be continued.)

Fig. 2.

CANAL AND RIVER NAVIGATION. BEADON'S WARPING SYSTEM.

Sir,-Several years ago, I obtained patents for the invention which I am about to describe, and afterwards made many attempts to get it introduced through railway engineers and directors, but did not succeed; it has not been published in its present form, and is therefore something new.

In the early stages of our railway system, good joints between rail and rail were considered by scientific men, and some railway engineers, as indispensable to public safety in railway conveyance. But somehow or other this feeling for railway perfection was soon abandoned, and the consequence has been, where chains and blocks are used for rails to rest upon, the joints in numerous places are now in a very bad condition, as represented by fig. 1; a a being the ends of two adjoining rails, and c the open joint between them.

I prevent the rails from separating by means of a solid link, having first prepared the ends of the rails to receive it, as shown by fig. 2; d is a plan of the link, and e an end view of it.

It will be seen that the opening formed at the ends of the rails, fig. 2, corresponds with the figure of the link, and, by having the circular edges of the link a little less in diameter than the diameter of the holes in the ends of the rails, fig. 2, sufficient allowance is afforded for the expansion and contraction of the rails.

Links of this description can be made with great facility from bars of iron rolled to the figure of the link, and afterwards cut by circular saws into the proper lengths the links should be. The rails, too, can be drilled and cut into at the ends with much expedition by machinery. I consider the expense of linking together a double line of rails, in the manner described, would not exceed 1657. per mile.

The advantage of a line of rails having close joints is very evident when we consider that 165 of them, on each side, are run over in the short space of one minute of time, if the carriages be travelling at the rate of 30 miles per hour. This being the case, so far as the senses of the passengers are concerned, they are never off the jolting notches during the whole length of their journey.

Besides this evil of giving passengers a good shaking, open joints indicate with perfect certainty that the rails have been shifted from their original position by

181

the tractive force of the locomotive engine-wheels. When this shifting takes place to a greater extent than shown by fig. 1, the last rail of the number moved will be off the chair. When the engine arrives at this point, down goes the end of the unsupported rail, jump goes the engine, and the chances are that next moment the whole train is off the line.

When such an accident takes place, and the train is travelling at a high velocity, we all know that the consequences are terrific. In nine cases out of ten, in my opinion, when locomotive engines run off the rails accidentally, it arises from the bad condition in which the rails are in at the joints. If this be the case, it proves an extraordinary fact, namely, that for the sake of saving 1657. per mile in the construction of the permanent way, the lives and limbs of the whole travelling population of Great Britain are placed in jeopardy. Steam having completely superseded land conveyance by coaches, the public have no alternative but to remain at home or follow locomotive engines wherever they go, which is not always a safe arrival at their termini. I am, Sir, yours, &c. JAMES WHITE, C. E. Manor-road, Clapham, Aug. 1846.

CANAL AND RIVER NAVIGATION.-CAP-
TAIN BEADON'S WARPING SYSTEM.

Some remarkably successful trials have been made this week, on the Regent's Canal, of a new system of propelling, patented by Captain Beadon. It consists in the combination of the power of steam with the mode of warping, so well known to all nautical men. In the middle of a flat-bottomed barge, with a prow and rudder at both ends, (Janus' fashion,) there is placed a steamengine, the power of which is transmitted to two rollers mounted in bearings, one at each end of the boat, which are alternately employed in letting out and drawing in a wire rope, either laid along the bottom of the canal, or carried from the boat to posts on the towing-path. In the experiments made this week, several heavily-laden barges were drawn by the steam-tug with ease at the rate of between five and six miles an hour. Of all the plans that have been recently brought out in aid of the declining canal interest, this seems by far the most promising. In our next we shall supply some further details.

ASHENHURST'S BOLT FOR MORTICE AND OTHER LOCKS.

stered under the Act for Protection of Articles of Utility. John Talbot Ashenhurst, of 41, Upper John-street, Fitzroy-square, cabinet and piano-forte maker, Proprietor.]

EFFICACIOUS RAZOR-STROP.