404

MECHANICAL GEOMETRY-MR. JOPLING'S SEPTENARY SYSTEM-MR. CHILD'S PARABOLIC TRAMMEL. Sir, I have already informed you that I have reason to think that all the lines of the first division are the epicycloids of the lines of the cusps; and now I believe I may say the same thing of almost every other division. Under the article "Epicycloids" in "Rees's Cyclopedia," it is stated, that "if a parabola be made to revolve upon another equal to it, its focus will describe a right line perpendicular to the axis of the quiescent parabola:" (see fig. 1.) Fig. 1.

"the vertex of the revolving parabola will also describe the cissoid of Diocles; and any other point of it will describe some one of Newton's defective hyperbolas having a double point in the like point of the quiescent parabola," And in the same work it is stated, that the" invention of the method of describing the cissoid by the continued motion of a square ruler is ascribed to Sir I. Newton." I cannot, however, find that it has hitherto been known, that the motion of a plane carried by the revolving parabola is identically the same as the

B

Fig. 3 shows two of these planes connected together, which exhibit at one view both the cissoidal motion and also the epicycloidal motion of the parabolas. As the elements for regulating the

motion are the same on each plane, it is, of course, not material to which the describing point is fixed. If the epicycloidal motion of the parabolas be only considered, the focus on B will describe the directing right line on A; and the focus on A the directing right line on B. Consequently, if a directing right line and a pole in the focus be fixed upon each plane, these may be substituted for the parabolas, and either plane will have the same motion. The several lines which may be generated by points upon either plane may be enumerated as follows:

First. The focus or pole describes a right line, which is uniformly symmetrical, and a line of the first order.

Second. The apex of the parabola, or a point on either plane midway between the pole and directing right line, will generate the cissoid, which is a cuspidated line in two parts symmetrical with infinite branches. This is a line of the third order.

Third. Any other point upon the periphery of the parabola, or line of the cusps, will produce an oblique or dissymmetrical cuspidated line. The farther the describing point is from the vertex, the greater the dissymmetry of the two brauches will be. These lines are called defective hyperbolas by Sir I. Newton.

Fourth. Any point upon the axis of the parabola, or that part of the symmetrical line within the parabola either above or below the focus, will draw an oblate symmetrical line.

Fifth. All other points within the parabola, or line of the cusps, will describe dissymmetrical oblate lines.

Sixth. Any point on the other part of the symmetrical line, or the produced axis of the parabola, will draw a symmetrical cuspidated line.

Seventh. And all other points of the plane beyond the parabola, or line of the cusps, will generate dissymmetrical nodated lines. The nodes will be the greater the farther the describing point is from the line of the cusps; and their dissymmetry the greater the farther the point is from the directing right line, and on that side of it which contains the line of

the cusps.

405

Or the enumeration may be thus briefly stated:

First. The right line. Second. The symmetrical cuspidated line.

Third. Dissymmetrical ditto lines. Fourth. Symmetrical oblate lines. Fifth. Dissymmetrical ditto. Sixth. Symmetrical nodated lines. And,

Seventh. Dissymmetrical do. do. Although the five last descriptions of lines are infinite in variety, those terms comprehend the whole; and I offer this as another proof of the possibility of forming a complete classification of all the lines in the septenary system.

Either plane may be supposed to be divided by the line of the cusps into two fields, both of which are infinite: but the one producing the oblate lines may be called the interior field, while the other which describes the nódated lines may be called the exterior field. And either plane is divided by the line of symmetricals into two equal and respectively similar parts.

It may perhaps be proper to state that any two points upon any line crossing perpendicularly the line of symmetricals, and at equal distances from it, will always produce lines respectively symmetrical, whether oblique, oblate, cuspidated, or nodated.

It may also be proper to observe, that while the epicycloids of the hyperbolas are finite lines, those of the parabola are infinite lines.

Portions of these lines, as well as those of many others, are beautifully described by an apparatus which I have called "The Nichomedean Rulers."

Mr. Child has had the kindness to send to me for inspection one of his parabolic trammels, which exhibits a proof of his early ingenuity. It is indeed the same in principle as the one of which you have published an engraving, and on which I ventured to make a few observations.

In both Mr. Child's trammels, the simple principle on which they are constructed is by the multiplicity of pieces not readily perceived. It is, however, simply as in

when compared with fig. 4, will show the difference between the conchoidal principle, on which Mr. Child's apparatus is constructed, and the cissoidal principle. In figure 4, the poles a is in the produced, directing right line c; and the pole bis at a distance from the directing right line d. In figure 7, both poles are at an equal distance from the respective directing right lines. In both cases, in whatever position the poles may be, perpendiculars to the directing right lines at those points will intersect in the periphery of the parabola. In fig. 7, the pole and focus of the parabola coincide. In fig. 4, the pole is at the apex, and the focus from the apex is of the distance between the pole b and the directing line d.

An enumeration of the lines produced by the conchoidal motion will be something like that for the cissoidal motion; but the reverse of the conchoids, or those produced by rolling B against A, fig. 6, as well as the intermediate cases between that and the cissoids, will require a very different classification.

Knowing that the ellipse and hyperbola were lines of the cusps in the first division, I ventured to state (vol. viii. p. 437), that I thought all the conic sections and the lines of the third order enumerated by Sir I.

DRURY'S

I am, Sir,

Your obedient servant, JOSEPH JOPLING.

IMPROVEMENTS IN BELL-
RINGING.

Sir, The communication of your correspondent C. H. in your last Number, has apprised me of an important omission in my description of Mr. Drury's improved mode of fixing and striking church-bells, at p. 282. I should there have stated, that the same principle can be applied to bells already hung with great facility, and Mr. Drury is prepared so to apply it.

As I before stated, many of our steeples are in such a dilapidated state, that it has become dangerous to ring the bells which they contain, and many others are fast approaching to"wards the same condition. In the first of these cases, it is absolutely necessary if the music of the bells is to be retained-that Mr. Drury's mode of ringing should be immediately adopted; in the latter, the same course is dictated, both by motives of safety and economy.

The parishioners of St. Mary-leBow were some time since in communication with Mr. Drury on the subject of their bells: and although they then declined having their peal recast upon the improved principle, yet they should certainly apply the new mode of striking to the bells, which now hang mute within their steeple the only course which will ensure the future safety of the edifice, with the continued harmony of the peal, to the delight of every true Cockney.

The mode of applying the new principle of striking to old bells, is tolerably well illustrated by fig. 1, p. 361.

407

The plan, fig. 2, same page, if adopted, would produce no better sound from the bell than the inferior one now obtained by clappering.

With regard to the "hope," expressed by your intelligent correspondent, that so important an improvement will not be suffered to expire in diagrams and theory"-I may observe, that Mr. Drury is a PRACTI CAL MAN, and one not likely to give up that, which, after so much trouble and expense, he has succeeded in bringing to perfection.

All, however, depends upon the PUBLIC; if they are willing to discontinue the present waste of human labour, and to remove the risk, which in so many cases now exists, of bring-, ing down the steeples about their ears -let them avail themselves of the ingenuity and skill which offer such a simple, inexpensive, and effectual remedy.

At some future time I hope to have the pleasure of laying before your readers some of the investigations of Mr. Drury, on a subject of great importance to the musical world. I am, Sir, Yours respectfully, WM. BADDELEY, Jun.

10