LONDON: Printed and Published by James Bounsall, at the Mechanics' Magazine Office, No. 166, Fleet-street.-Sold by A. and W. Galignani, Rue Vivienne, Paris; Machin and Co., Dublin; and W. C. Campbell and Co., Hamburgh,

Mechanics' Magazine,

MUSEUM, REGISTER, JOURNAL, AND GAZETTE.

MR. DE LA HAYE'S PLAN OF A SUBMARINE RAILWAY.

RESPECTED FRIEND,-Having thought for some years, that submarine railways might be constructed in rivers and nårrow seas, I take the liberty of intruding on thy notice the following plan, by which I consider this object may be attained. The plan is to construct wroughtiron tunnels, and then to place them on the bed of the sea.

I beg to state, that according to a proImise which I received from Sir Joshua Walmsley, to whom I had submitted my plan, a copy of it must have been in the hands of George Stephenson at least ten months; perhaps that engineer will not refuse to confess, that his plan for the tunnel over the Menai Straits was copied word for word from my plan, with the only modification of suspending it above the water.

I would construct wrought-iron tunnels in divisions of about 400 feet in length, by making a frame, or rather skeleton work, of strong wrought-iron bars, or ribs, in such a manner as to admit of three distinct sets of iron plates being riveted to it; the space between the plates I would fill with a mixture of tar, pitch, and sawdust, so that even if the outer plate received any injury, the water could not enter the tunnel.

The whole of the divisions being built, I would block up both ends with a temporary framework of wood and iron; then I would launch them at sea, and tow them to the spot by means of a steam vessel. I would then fill them with water by the stop-cocks, so as to cause them to sink on the bed of the water; the divisions should then be temporarily connected outside from diving bells, then the water pumped from inside, the temporary blockading frames removed, and iron plates screwed between the divisions, so as to render the whole as one vast building.

To prevent the tunnels from rising by their buoyancy, they should be constructed with a wide platform on each side, as represented in figure 1.

A number of very large stones should be let down on the platform on each side, in sufficient number to keep the tunnel in its position; these stones should of course be placed below, previous to pumping the water from the inside.

Supposing it were possible to construct

such a tunnel between Dover and Calais, it would be necessary to sink the divisions near the shore below the bed of the water, by forming a kind of canal for their reception; this would protect the building from the violence of the waves while in deep water, as it has been ascertained, I believe, that the sea is not agitated in great depths, even in the most violent storms. I believe that the bed of the channel is sufficiently level to admit of such a building being placed on it. I am aware, however, that the expense would be very considerable, though probably much less than if a tunnel were bored under the bed of the channel, and attended with very little danger.

In rivers it would be necessary to sink a great part of the tunnel below the bed of the water, because the centre of rivers is considerably deeper than near the sides; and in fact, by placing nearly the whole under ground it would offer no impediment to the navigation of the river.

Wrought-iron tunnels might also be used instead of suspension bridges, and would undoubtedly be much stronger. I would not, however, place the rails inside, but on the top. I would construct such a tunnel in the form of a cylinder, as represented in fig. 2.

Over the cylinder I would place a thick wrought-iron platform, supported on castiron mountings, so as to admit the wind passing round the cylinder; by these means the building would not be liable to suffer injury by the strongest gale. I would also place a number of cast-iron columns inside, to support the upper arch and platforms.

Smaller tunnels might be made for wider channels; these might be made of cast-iron, as they would not require being more than 3 or 4 feet diameter; rails might be placed inside, on which a small car for the mails could be drawn by a stationary engine at each end; wires might also be laid inside, for communication by means of the electric telegraph.

Another use for iron tunnels would be in parks and pleasure-grounds. Where a railway passing through destroys, in some cases, nearly half the value of the estate, an iron tunnel might be sunk under, if it were even at the inconvenience of having a considerable gradient at each

SOLUTION OF A MECHANICAL PROBLEM.

end; or in other cases, the tunnel might be only partly sunk in the ground, and the other part covered with earth, or both sides, in a sloping direction.

In fact, I consider that the modifications which this invention admits of are almost infinite.

I have now stretched my remarks to too great a length to admit of my entering into any further details at present. If thou wilt permit, I will in a future communication place my plans for sinking the bed of the water, for placing the tunnels beyond the reach of injury, and also the mode of connecting the divisions under water.

I remain respectfully,

JOHN DE LA HAYE.

8 Mo. 18th, 1845. 100, London-road, Liverpool.

SOLUTION OF A MECHANICAL PROBLEM.

PROBLEM.-In a certain arrangement of machinery, there are three pinions of the same pitch, but of different diameters, all moving on independent axes, which are parallel to one another, and working individually or collectively into a spur-wheel of equal pitch, moving also on an independent axis parallel to the former; it is required to determine the size and position of the spur-wheel, the diameters and distances between the centres of the pinions being severally given.

It is easy to perceive that the object of this arrangement is to obtain three independent motions, either separately or simultaneously, according to circumstances; but the mode of solution, as the problem is here enunciated, is not very obvious; it will, therefore, be better to divest it of its mechanical technicalities, and state it in general terms, as follows:- To find a point such that straight lines drawn therefrom, to three given points in the same plane with it, shall have given differences. Or, other wise, thus:- To find a point in a given right-lined triangle, from which, if straight lines be drawn to the angular points, the differences of these lines shall be given.

It will readily be admitted that the problem as thus generalized is the same as that proposed, the condition of different diameters being involved in the differences of the lines; but under this modi

163

fication the obscurity disappears, and, in consequence, the method of solution is at once made manifest. We shall, in the first place, endeavour to resolve the problem algebraically; we are aware, à priori, that the equation, when brought to its ultimate form, will assume a very complicated aspect, by reason of the number of terms that it must necessarily involve; but when applied to the solution of a specific example, the complexity will in a great measure disappear; as in that case the several symbols will be replaced by their equivalent numbers and the equation will be reduced to form ready for solution. The operation in this way is as follows:

Let A B C be the plane triangle, of which the sides A B, A C, and B C, are given, and let P be the position of the point required; then, by the problem,

B

B

the point P must be so placed, that the straight lines P A, PB, and P C, drawn from it to the angular points A, B, and C, shall have given differences.

From the vertex C, upon the base A B, let fall the perpendicular C D, dividing A B into the segments A D and B D, whose values can be determined; and from the point P let fall the perpendiculars PQ and P R, meeting the base A B in Q, and the perpendicular CD in R; then we have formed the rightangled triangled triangles A QP, BQP, and P R C, having the required lines PA, P B, and P C, for their hypotheneuses respectively, P B being assumed the greatest. From this construction, the method of forming the three independent equations by which the position of the point P is to be determined, is obvious.

=

Put a A D, one segment of the base made by the perpendicular C D,

b=B D, the other segment, c=CD, the perpendicular from the vertex on the base A B,

d=PB-P A, the difference between the lines drawn from P to the angles A and C,

=P B-P C, the difference between the lines drawn from P to the angles B and C.

These are the several data concerned in the problem, and which are either directly given, or can be found from the figure by well-known geometrical principles; and the following are the incognita, or things unknown, viz.,

x=D Q, the distance between the perpendiculars CD and P Q, from the points P and C.

1. B P2=B Q2-P Q2; that is

2. A P2 A Q2-P Q2; that is

=

3. C P2=C R2-P R2; that is Now, since we have assumed the straight line P B to be greater than either PA or PC, it follows that the first equation is greater than either the second or third; for this reason, let each of them be subtracted from the first, and we shall have the two following remainders, viz.,

4. 2dz-d2= b2 — a2 + 2bx+2ax. 5. 28z- &2= b2 - c2 + 2bx+2cy.

y=

...

y=P Q, the perpendicular distance between the base A B and the required point P.

z=B P; the distance between the angle at B and the required point P, being greater than either of the other two distances, PA, or P C.

Then we have

A Q=A D-D Q=a−x ;
BQ=BD+DQ=b+x;
CR=DC-DR=c-y;
PA=PB-d=z-d;

And P C P B→=z-ô;
therefore, by the property of the right-
angled triangle, we get the following
equations, viz.,-

z2=b2+2bx+x2 + y2.

z2 - 2dz + d2= a2 −2ax + x2 + y2.
z2—2dz+d2=c2 − 2cy + x2 + y2.

From the fourth equation, by transposition and division, we get

13 y =

;

; then

and, in like manner, from the fifth, we
28z −2bx + c2 — b2—d2
have
2 c
for x, in this latter expression, substitute
its value in the former, and we obtain

(a+b) (2dz + c2 − b2 − 82) − b (2dz + a2 − b2 — d2)
2c (a+b)

Let the square of this value of y, together
with the values of x and x2, as obtained
above, be substituted instead of them
b (2dz + a2-b2 — d2)
(a + b)

respectively in No. 1, and we shall obtain the following equation, involving only z and known quantities :

+

2dz + a2-b2 — d2
2 (a+b)

— b (2dz + a2 — b2— d2) } 2.

2c(a+b)