more than the stone. In an arch where stone- and brickwork are combined, little reliance should be placed on their connexion, as this is always more or less disturbed after the centring is removed, so that we should endeavour to construct each portion of the arch with its bearing surfaces or beds as nearly equal as possible.

In the first models the soffits of all the stones were made of an equal length, considering that this would present the best appearance; but this method rendered the bearing surfaces very unequal, as will be seen by fig. 9; the equal lengths being indicated by the dotted lines.

This difficulty is overcome by this simple means: instead of having the stones of equal length on the soffit, they are made so on the intermediate development, and then the areas of the bearing surfaces or beds of the stones are all equal. See fig. 10.

Having given the mode of laying out the lines, I will now proceed to the practical part, viz. the working of the individual stones.

My first idea was to commence by working the soffit; and this was the mode employed.

Having obtained an elastic mould cut to the angle at which the joints of the soffit cross the axis of the bridge, the workman by means of this gets an oblique line on that surface of the stone which he intends for the soffit. It will be understood from fig. 11, that this oblique line thus obtained will be parallel with the axis of the bridge. The workman then proceeds to chisel out a groove (or what is by masons called a chisel-draught) along this line, of sufficient depth for what he knows will be required for the hollowing of the stone.

He then takes two wooden moulds (one of which is shown in fig. 12), which are portions of the same circle as the soffit itself. A mark being placed upon the centre of each of these moulds, the workman then proceeds to sink them into the stones at right angles to this chisel-draught, (see fig. 11,) and in such a manner that the centre marks shall be in the chiseldraught, and the upper edges of the moulds, which are straight, shall be in the same plane, or what is commonly called, out of winding. It will now be obvious that these two last grooves will form true portions of the soffit itself, and therefore, that the workman has nothing to do but to work out the remainder of the stone with a straight edge, always kept parallel with the first draught, and sunk to the bottom of the two draughts which were worked by the curved moulds. Having obtained this hollowed surface, an elastic mould, of the exact size of the soffit of each stone, is pressed into it, by which

the stone being marked, we obtain all the lines of the soffit itself.

It will now be quite evident that the beds may be obtained by making use of a square, one limb of which shall be made to the curvature of the soffit, and the other the radius of this curve; always taking care that this square is kept at right angles to the axis, as will be seen in figures 13, 14, and 15.

The first few stones were wrought in this manner;. but finding it very difficult to prevent the workman from getting his soffit a little on one side, by which means he wasted much of the stone on one bed and rendered the other deficient, I had recourse to a method which I will describe. Having provided two straight edges, the one parallel and the other containing the angle of the twist, (see fig. 16,) we proceeded to work one of the beds by chiselling two draughts along the stone, so that these straight edges being kept at a proper distance from each other were let into the stone until they were out of winding on their upper edges.

Having finished one bed by straight edges, we then obtained the soffits and other beds by means of the square before mentioned. By working a bed first instead of the soffit, the best will always be made of a block of stone.

As we have before seen that all the stones constituting a skew arch are portions of the same square threaded screw, the workman having finished one stone has only to repeat the same operations with every other.

Any stone in the face of the arch, taken from one side, and applied to the corresponding one face to face, will continue the true spiral plane: this fact enabled us to work all the stones for one bridge in pairs; that is, one stone having been wrought with the proper twist, and of sufficient length to make two stones, was accordingly sawn in two at the proper angle: but of course this cannot be done advantageously when the stone is of a very hard nature.

It has been shown that by developing all the various surfaces, instead of having to think of complicated spiral lines, they are at once reduced to straight ones; and I will now very briefly show how simply the data necessary for the construction of a skew arch may be obtained (see fig. 17).

Let A represent the curvature of the intrados, and C the extrados, B being a line midway between A and C. Let DD, EE, FF represent the boundaries of three cylinders of which A, B, C are the transverse sections; let these cylinders be cut by the straight line G, H, at the angle of askew, that is, the angle formed by the two roads crossing each other; and from the points I, J, K, draw three straight lines at right

angles to the axis, and of such lengths that I L shall be of equal length to the semicircle A, and J M equal to B, and K N equal to C; from the point O draw the straight line O L, and also from P to M: it will be seen that OL is the approximate line of the developed soffit, and PM that of the intermediate development. Add Q, R, and S, which are the centre lines of the three developments.

It will be seen that when these developments are placed as in an arch, these three lines Q, R, S being parallel with the axis, will be in a plane perpendicular to the axis, and, therefore, that all the points in each spiral will be vertical with the axis, and also with one another.

Through any point in P M draw a straight line V at right angles with PM, which straight line shall extend to the axis of the cylinder.

At the point where it intersects R, a line T perpendicular to the axis intersects R also: this last perpendicular line cuts the three lines Q, R, S at the points where the lines U, V, W, which meet in X, intersect Q, R, S.

The joints are then drawn upon the three developments parallel with the lines U, V, W, and at such distances that the lines Q, R, S shall be cut into equal parts. Of course, care must be taken to divide the approximate line of the soffit into a given number of stones. The angle X will be that which the intrados form with the axis of the cylinder, and the angle UW will give the wind of the bed. On this principle and by the rules here given, it is nearly as easy to work the stones of a skew bridge as those of any other.

Park Village East, London, March 17, 1836.

LXIII. Further Observations on M. Cauchy's Theory of the Dispersion of Light. By the Rev. BADEN POWELL, M.A., F.R.S., Savilian Professor of Geometry, Oxford.

I

(Continued from p. 28.)

PROCEED to illustrate the further researches to which I alluded in my last paper; relative to the development of the theory of dispersion, and simplifying the process of M. Cauchy.

In order to consider the subject in its simplest form, let us confine our attention to a plane wave perpendicular to the axis of x, with vibrations parallel to the axis of y. Then the displacements & and will vanish, and the differential equation

of motion deduced upon M. Cauchy's principle (in my analysis, eq. (12.),) will be reduced to

n

where is the value, at the time t, of the varying displacement of the molecule m, whose rectangular coordinates when in equilibrium are xyz;+An is the displacement at the same moment t, of another molecule m, which has for its rectangular coordinates when in equilibrium

or the distance between these two molecules in their positions of equilibrium; B is the angle between this distance and the axis of y; and, finally, f(r) and ƒ (r) are functions of r, of which the former (if positive) expresses the law of attraction, or (if negative) the law of repulsion, and the latter is derived from it by the rule

ƒ (r) = r f' (r) — f (r).

S, the sign of summation, is relative to the actions (attractive or repulsive) of all the molecules m.

I have recapitulated thus far in reference to what was established at the outset of M. Cauchy's investigations. Now this analysis is thus far devoid of all difficulty or intricacy; the whole difficulty of the subject lies in the integration of these equations of motion. The integration given by M. Cauchy is of an extremely general kind: but for the purpose we have now more immediately in view, it will be readily allowed that if a particular solution were proposed, such as to include the establishment of the relation between μ and A, it would suffice. A valuable instance of a method of effecting such a simplification has been laid before the readers of this Journal, in the excellent paper of Mr. Tovey in the Number for January, p. 7. But another such particular solution has been pointed out by Sir W. R. Hamilton, the nature of which I now proceed to describe; and this will be most perspicuously done in the following manner:

It will be easily seen that all the conditions of a wave for the ordinary phænomena are fulfilled by such a function as

which, merely by the assumption of the coefficients and trigonometrical operations, is easily put under the form

to being entirely arbitrary, and 。, being also arbitrary, but small; is introduced only for greater generality.

10

Differentiating in respect of t, we shall have

2

- (2)2 %, cos (27 (μx-t+to)). (4.)

T

T

Also, by the method of finite differences, we have

Now (on precisely the same grounds as those adverted to in the analysis of Cauchy for deducing the equations (22.),) it will be seen that this expression is of such a form that if it were introduced in a summation, since we may assume half the values of Ar as positive and half as negative, the second member involving the first power of the sine of a function of ▲r, and the first member the square, the sums of all the values in the second member will destroy each other, but not those in the first.

Thus on substituting this value of A in the differential equation (12.), or that above, (1.), we shall only have to take into account the first member, multiplied by the function of (r); and it will thus easily appear that that equation (1.) is satisfied by these values derived from the assumed equation of the wave (3.), provided we suppose

2

πμ Δε

(==) ' = 8 {2m f(r) + cos' Af(r) (sin x) }; (6.)

s

r

T

or, in other words, the equation (3.) coupled with this last condition (6.) is a particular solution of the differential equation of the motion of a system of molecules (1.).

But also, this equation (6.) involves the relation between

and μ, (or between λ and μ, since we have

is expressed by writing, for abridgement,

- =

T

1

which