from the overhead motion. The carrying plate c of the instrument is attached to this pulley, and is of course driven by it.

The stem is bored through, and a small shaft put inside it, having at one end the central wheel d, and at the other a worm-wheel of 150 teeth, geared into by a tangent screw carried by a frame fixed to the end of the square stem of the instrument. The wheel is also graduated on its edge into 72 divisions read off from a pointer on the frame carrying the tangent screw.

Attached to the carrier plate c is a radial arm e having on it a boss bored through, and fitted with the stem or spindle of the eccentric arm f. This arm is precisely similar in construction to the arm of the eccentric cutting instrument. At the other end of the stem or spindle of the eccentric arm, and inside the radial arm e, there is a brass pinion g, geared into by the wheel h, which, in turn, is driven by the central wheel d. The radial arm is fastened to the carrier-plate at two points, by the screw i and the screw k. The former is the centre round which the arm radiates, and is also the spindle or axis upon which the intermediate wheels revolve; the other screw goes through a curved slot in the arm, as will be seen. The arm can therefore be moved around on the centre screw i, to the extent of the slot, and is fastened at any point of this movement by tightening the milled head screw k. This movement of the arm is effected by means of the small screw the end and collar of which is seen, the square extent of movement being indicated by the graduated divisions marked on the edge of the instrument. These divisions are such that a movement of one will throw

on c,

the centre of the eccentric spindle th of an inch out

of place.

Now when this instrument is in the slide-rest and is set in motion by a gut from overhead passing round the pulley b, the radial arm being fixed at the zero point of the graduated scale and the point of the tool adjusted central, a dot only will be produced. If, however, we stop the instrument, and turn round the eccentric screw to move the tool away from the centre, and again set the instrument in motion, a circle will be cut, the radius of which will depend upon the number of turns the eccentric screw was moved.

By moving the screw of the slide-rest we can place a series of circles in a row of any length and at any distance apart, and by using the index-plate of the lathe-spindle we can arrange these circles anywhere round a centre, and at any distance from that centre. It will thus be seen that the instrument, when thus only used, has exactly the same power as the eccentric cutting instrument.

This instrument is also adjusted central with the little knot left on the surface to be figured, in the same manner as was explained for the eccentric instrument. Before going further we will suppose the tool-point to have been thus adjusted central. If the screw k be loosed, and the radial arm shifted say 20 divisions of the graduated scale, which will throw the centre of the cutter-spindle ths of an inch out of place, and the tool also shifted 20 divisions or ths of an inch along the eccentric slide, on the instrument being set in motion and applied to the surface, the tool will travel across the centre of the work and produce a

straight lineths of an inch long, ths each side of the centre.

If the eccentricity of the radial arm be now reduced 10 divisions, which will leave it centre, on again putting it into

th of an inch out of motion and applying

the tool to the work, an ellipse will be produced with a conjugate or short diameter of ths of an inch, and a transverse diameter ofths of an inch in length.

By altering the ratio between the eccentricity of the radial arm and the cutting tool, an ellipse of any proportions can be produced. So far as the shape of the figure is concerned it makes no difference whether the eccentricity of the cutter is more than that of the radial arm, or vice versâ; but in practice the former is to be preferred, as the cutting edge of the tool is presented to its cut in a more favourable manner.

The eccentricity of the cutter axis is caused by the movement of the arm e, and this movement being around the centre i, the cutter axis is evidently moved in the arc of a circle and around the wheel h.

If the radial arm be moved out 10 divisions and the cutter be moved 10 also, on the tool being put in motion and applied to cut, a straight line will be produced, and if we now shift both 10 divisions more, a straight line double the former length will be produced; but this line will not come on the same place as the last, nor parallel with it, but at an acute angle to it. We now find the use of the tangent screw at the end of the stem a; this is so graduated that, by moving it the same number of divisions as the flange, the radial movement of the cutter axis is counteracted.

We are indebted to Mr. Perigal for this neat improvement on Captain Ash's ellipse instrument.

When this instrument has cut an ellipse, any number of these figures can be arranged in a straight line by shifting the slide-rest, and if the centre of the instrument coincide with the lathe-centre, the ellipses may be placed to cut across each other in any angular position by moving the division-plate of the lathe. When, however, the two centres of the lathe and instrument do not coincide, the movement of the latheplate has the effect of arranging the ellipses eccentric to a common centre; and in this case the angular position of the ellipses is determined by moving round the tangent screw at the end of the instrument. The

worm-wheel having 150 teeth, if the tangent screw be moved round 37 times, the ellipse will be struck at right angles to its former position. Any smaller angle can be obtained by moving the tangent screw a less number of turns; the movement is generally better indicated by the division index on the edge of the wheel. Fig. 194 is formed by a series of vertical ellipses arranged round a centre by the movement of the division-plate of the lathe. Fig. 195 is formed of

a series of horizontal ellipses arranged round a centre by the same means.

It will be well to bear in mind that, in using this instrument, when the radial arm and the cutting instrument are both moved out of centre the same distance, the tool will cut a straight line.

In order to produce an ellipse there must be some difference between the eccentricities, and the proportions of the ellipse will depend on the extent of the

two movements.

The transverse or long diameter will be equal to double the sum of the two eccentricities, and the conjugate diameter will be equal to double their difference; the straight line may therefore be looked upon as an ellipse with a conjugate diameter=0. This instrument cuts an ellipse when the eccentric cutter makes two revolutions, whilst the pulley and carrier-plate make one in the opposite direction. In this case there will be a 48-tooth wheel as a stationary driver, gearing with a wheel of 24 teeth on the same socket with a 36-tooth wheel gearing with another 36 on the end of the eccentric frame spindle. When the instrument is in motion the 48-tooth wheel is still stationary and the 24 is carried round it, making two turns every time, and these two turns are coinmunicated to the eccentric spindle by the two 36-tooth wheels.

If we remove these two 36-tooth wheels, and put a wheel of 48 teeth on the socket with the 24-tooth wheel already on, and put a 24-tooth wheel on the end of the eccentric spindle, on putting the instrument in motion the cutting frame will no longer make only two rotations whilst the carrier-plate and pulley make one, but it will