TROCHOIDAL CUli 



TROl'HV. 



curve b> in apocentral or perieentral radii, or in the radii opposite to 

 them. Substitute for y and jc their value* in jr tan kp.x, and this 

 quatiun is eoiilyireduced to 



a sin O - i>) + i sin (* {> In) = ; 



from which the values of <> which belong to double points answer 

 ing to different integer values of i are to be obtained by approxi- 

 mation. 



Tho radius of curvature of any trochoidal curve is obtained from the 

 formula.' (9, 10), and is as follows : 



{<i ; + 4 J * + 2oi)i cos (n 



i : H 5 + 04 (11= + H) COd (ll - 1) * 



It never becomes nothing except at tho cusps of the epicycloid or 

 hypocycloid ; nor infinite except at the points of contrary flexure, or 

 at the pericentre in the case of great approach to straightneiw already 

 alluded to. The equation of the Evolute [INVOLUTE AND EVOLUTIJ, 

 { and i) being co-ordinates of a point in it, is involved in the following 

 equations : 



H = a (1 - ) sin ^ + 6 (1 - n ) cos $ 

 { = a (1 ) cos $> + 4 (1 nit) cos n<t> 



o* + 6**. + 2a&cos ( 1) 

 bej" 8 O t + j B * + ,,6 ( + ) cos 1) f 



The evolute, then, of a trochoidal curve may be described as of an 

 extended sort of planetary character, having on epicycle of variable 

 size, and radius 6(1 nit), which moves upon a deferent, also of variable 

 size, whose radios is a (1 it). In one remarkable pair of cases this 

 variation of the elements of the evolute ceases, namely, for the epicy- 

 cloid and hypocycloid. If tfn'a", we find that K becomes the con- 

 stant 2-^(1 +n), and the equations of the evolute become those of a 

 new epicycloid or hypocycloid, having its vertices at the cusps of the 

 original involute. 



The arc of the trochoidal curve cannot be obtained without tho 

 previous rectification of an ellipse, except in the case of an hypocycloid 

 or epicycloid. In the former case the arc measured from the apo- 

 ceutre or cusp, formed while the deferential angle <p is produced, is 



4ab a + b 



and the whole arc of descent is iab~- (a + b), 

 epicycloid, tho apoceutre being now a vertex, is 



4ab 0-6 



Tho same for the 



the whole arc of descent being 4ab -i- (o b). All the preceding 

 formula; may be reduced to those of the trochoidal form by the 

 general equations before given. 



The equations of the cycloid and simple trochoids may be, in on 

 article like the present, best deduced from those of the double circular 

 system. Remove the origin from the centre to the circumference of 

 the deferent, we have then for the equations 



:r= o(l cos0) + <icosn, y=a HJnf + isin n<t>. 



Let a Increase without limit, and let t be the arc of the deferent, 

 described with a velocity o, while y is the absolute angular velocity of 

 the planet round the epicycle. We have then i = a<f>, and a : > : : v : nip. 

 Substitution gives 



ar o(l cos-^+ icos-*, y=osin + &sin-i; 



of which, when a is increased without limit, tho ultimate equations 

 arc 



<r=6cos i, . y= 



which are the equations of the common trochoidal system. Hero an 

 epicycle moves up the axis of y with a velocity o", while a plam-t 

 revolves in the epicycle with tho angular velocity v. When rb in 

 numerically greater than a, there is a curve with loops ; when v&a, 

 one with cups ; and when 6 U less than o, one with undulations and 

 contrary flexures. 



To get the equations of the other extreme case, we must reduce 

 the planetary form of the general equations to the trochoidal, which 

 f 



*- (r + B) cos f+ coos ( - ;- 

 -- 



Let c =n-H : that is, let n lie the distance of the jilnnct from the 

 circumference in contact of tho rolling circle. Substitute f'>r c, 

 develop 



:'. , 



(*!) 



- : . I :: 



into their binomial forms, and then increase the rolling circle without 

 limit. Thu ultimate equations will bo found to bo 



* (r + n) cos <t> + f<t> fin ^ 



Here a tangent rolls over the fixed circle, carrying with it a point 

 attached by a perpendicular arm of the length II. If 11 = 0, we have 

 the involute of the ftxc.l rircle: hut if 11= r, in which oase the 

 moving point begins from the centre of the fixed circle, we can deduce 

 rr+, ? |w, which gives the spiral of Archimedes. We have 

 touched Hlightly on these extreme cases, but the mathematical student 

 will find it a useful exercise to develop them more fully. 



In conclusion, we wish to call attention to the fact that the speci- 

 mens of curves in this article are all entirely the production of 

 machinery. We are indebted for them to the kindness of Mr. Henry 

 Perigal, a gentleman who practises the higher branches of turning as an 

 amateur, and has devoted much time and money to the investigation 

 of the effects of double and treble motions, as shown in various 

 interesting publications. Most of those given in this article were 

 executed in his lathe by means of Ibbetson's geometric chuck, a 

 contrivance the results of which are well known to turners, but which 

 have never been exhibited, as far as wo know, in any article professing 

 to give /a mathematical classification of them. 



The preceding is the curve called the trisectrix in the article Tm- 

 ' : it is a wood-cut made in the usual way from a drawing made 

 iu tin lathe ; all the rest are cut in the lathe. 



TKO.M 1 '' ' N K ^ Italian, great trumptl). This ancient instrument was 

 formerly known in England under the name of aacbut, from the old 

 French taqiubuU. It is a deep-toned trumpet, composed of sliding 

 tubes, by means of which every sound in the diatonic and chromatic 

 scales, being within its compass, is obtained in perfect tune. The 

 trombone is of three kinds, the alto, the trnnr, and the bate, and 

 these, in orchestral music, are generally used together, forming a com- 

 plete harmony in themselves. 



The scale of the ulto-trmnhnnr is from c, the second space in the base, 

 to o, an octave above the treble clef :< 



' 



Tho scale of the tenor-trombone is from n, the second line in the basa 

 to A, thu second space in thu treble : 



The scale of the Ixur-lrambone Is from o, an octave below the second 

 space in the base, to o, tho second line in the treble : 



The froml-onc, when judiciously employed, a, for inntanco, In 

 M"/iut'* ' !iiv|iiii'iu,' and in hi- ' aiini/ U most <! 



jirodueing great and mil , M; but, by the follower* of (lie 



iltra-modorn school, its power is exceedingly abused, especially in 

 Italian o|H>ran. 



'J'Hof'IIV (rpiirtuor, tropacum) contains the same root as the Greek 

 verb Tptxta', " to turn or to put to flight," and was therefore originally 

 a sign or memorial cn-cU-il ..M the *|><>t. where an enemy had IKM-M eon- 

 I or put to flight. The custom of erecting such memorials of 

 victories, either on the lieM of Kittl<: or in the capital oi the conquering 

 i.itii ni, has been more or less common to all nations from the 1 

 remote to the most modern times. It was most general among the 



