TROCHOIDAL CURVES. 



TROCHOIDAL CURVES. 



also be included the extreme cue in which one of the 

 motion^ ig rectilinear, which give* the common trochoid, the cycloid, 

 and m clan of spirals which include* the involute of the circle, the 

 spiral of Archimedes, and other*. 



There are two ways of considering the** curve*. The first, which is 

 universally adopted, may properly be called the trofhoidal mode 

 (rpoxof, a hoop), because in it one circle is made to roll like a hoop, 



either upon a straight line, or upon the circumference of another 

 circle. The second, which we believe might be advantageously sub- 

 stituted for the first, we shall propose to call the planetary mode, 

 because it resembles the consideration of the manner in which a planet 

 anil it* satellite move round the sun. Here a circle, without any 

 rolling, has its centre carried round the circumference of another. As 

 there is no elementary work which treats of these combined motions, 

 though come understanding of them is necessary even for the purposes 

 of the most elementary astronomy, we shall first enter into this subject 

 at more length than usual, endeavouring to make ourselves understood 

 by those who have the first notions of geometry and of the composition 

 of motion : we shall then, more briefly, consider the application of the 

 differential calculus. 

 Let the point ti (fy. 1) be carried uniformly round the circum- 



Fig. 1. 



ference of the circle A M, and let M be the centre of a circle whose 

 radius is Me (AB). Let a point p be carried about the moving circle, 

 so that its angular velocity from a line of fixed direction in the moving 

 circle (say M c parallel to A B) always bears a given proportion to the 

 angular velocity of M, say that of : 1 ; that is, when the line o M has 

 described the angle A o M, the line M r has described the angle CMP, 

 which is it times A o M. It has been supposed that when ii was at A, P 

 was at B. The point r will describe a curve which is one of those called 

 Irocknidal ; or, on this explanation, planetary. And the circle c P being 

 always contained between two fixed circles, B E and b e, the planetary 

 curve is always contained between those two circles. We shall now 

 propose a nomenclature for the principal parts of this system. 



As in the Ptolemaic mode of considering the planets, let the fixed 

 circle A 1C be called the Afferent, the moving circle c P the epicycle. Let 

 M be the mean point, T the planet, o the centre, and let the planet be 

 >a!d to be in its apocentre or perictntre, when it is farthest from, or 

 nearest to, the centre. And as every apocentre must lie on B E, and 

 very perioentre on be, let these circles be called respectively apocentral 

 and pcriocntraL Let o P, as usual, be called the radius of the curve, or 

 iU radix* vector ; and the angle A O P its rectorial angle. When the 

 revolution is in the direction from A to M, let it be called direct ; when 

 in the contrary direction, retrograde. Let angle A o M be called the 

 mean or deferential angle, and denoted by <f> ; and let CMP be called the 

 epicydic angle, being denoted of course by n $>. Let the angle con- 

 tained between the pericentral and apocentral radii be called the anylt 

 of descent The following theorems will be readily seen, as soon as 

 these terms are understood : 



1. The planetary curve beginning from its apocentre at B, and the 

 epicydic motion being direct, and greater than the mean motion, there 

 will be a pericentre as soon a* on P has gained two right angles upon 

 CHE or A on, that is, when $>-$ = ISO*, or ^ is ISO' divided 

 by-l. 



2. But if the epicyclic angular motion, being still direct, be lew than 

 the mean motion, so that c n E is greater than CMP, there will ). ,1 

 pericentro when c M E has gained two right angles upon c M p, or when 

 *>-*= 180', or * is 180' divided by (1-n). 



3. And if the epicyclic motion be retrograde, so that P begins to 

 move the other way from o, there will be a pericentre when c M E and 



CMP together make two right angles, or when f+k+xlM*, or when 

 * is 1 80* divided by 1 + n, 



4. When the planet has come to its pericentre, it will begin imme- 

 diately to ascend towards the next apocentre, in a curve of the same 

 form as that by which it descended, but inverted in position, the parU 

 preceding and following the perioentre being alike, and the part pre- 

 ceding the next apocentre resembling that following the last apocentre. 

 As soon as the second apocentre is gained, the curve will start again, 

 in the same manner as the first. If N be a commensurable number, 

 say 7>-=-o, where p and q are integers, and the fraction be in its lowest 

 terms, the curve will return into itself when M has completed q revo- 

 lutions: there will be, if the epicycle be direct, p q orqp apooentres, 

 and as many perioentres ; but if the epicycle be retrograde, there will 

 be p + q apocentres, and as many pericentres. But if n be incommen- 

 surable, the convolutions will go on for ever, and the curve will never 

 be completed. We have therefore, in order to obtain the form of the 

 curve, only to consider one descent from apocentre to pericentre, or 

 one ascent from pericentre to apocentre: though the general appearance 

 of the curve depends much on the effect of many convolutions. 



5. Every planetary curve may be described by two distinct epicyclic 

 motions. For if we describe the parallelogram o M p q, we see that the 

 point q describes a fixed circle equal to the epicycle, while q v is t In- 

 radius of a moving circle equal to the deferent. If then the radius of 

 the deferent be a, that of the epicycle b, and the epicyclic angular 

 velocity be n times the deferential or mean velocity, it gives the same 

 planetary curve as if the radius of the deferent were b, that of tin- 

 epicycle a, and the epicyclic velocity 1 :th of the deferential. If, 

 then, we take the epicycle to be the least of the two, we do not limit 

 our investigation, provided we consider every possible case of epicyclic 

 velocity. 



The actual motion of the planet, compounded of both motions, 

 deferential and epicyclic, may be either direct or retrograde ; and the 

 curves may be best classified by observing whether the motions at the 

 pericentres and apocentres are direct, retrograde, or neither. At 1 

 is represented a case of each motion, pericentral and apocentral, 

 direct ; at 2, a case of each motion, when it is neither direct nor retro- 

 grade that U, directly towards the centre ; at 3, a case of each motion, 

 retrograde. We shall now consider how to make the separation of 

 these cases. 



First, as to the apocentres. When the motion in the epicycle is 

 direct ( for abbreviation say when the epicycle is direct), the two 

 motions conspire ; the line M c, then at A B, is being carried forward 

 with the velocity of A, while P is being carried from M c. Let angles 

 be measured in theoretical unite [ANCLE], and let the deferential or 

 mean velocity be 1, then the linear velocity of A is a, and that of p, 

 when at B, is the linear velocity to an angular velocity n and radius 4, 

 or n b. Consequently, a + n b is the apocentral velocity, which is direct ; 

 or the apocentral velocity is always direct when the epicycle is direct. 

 But if the epicycle be retrograde, the line M o is advancing with the 

 velocity a, while p is receding from it with the velocity no. Conse- 

 quently, when the epicycle is retrograde, the apocentral velocity is 

 direct, neither, or retrograde, according as a is greater than, equal to, 

 or less than, n b. 



Next, as to the pericentres. We can make a pericentre by supposing 

 the planet to be at 6 when M c is on A B. Now, if the epicycle be 

 direct [MOTION, DIRKCTIOX OK], the line B b being carried forward with 

 the velocity a, the planet is carried backwards with the velocity nli. 

 Consequently, when the epicycle is direct, the pericentral motion in 

 direct, neither, or retrograde, according as a is greater than, equal to, 

 or less than, n b. But when the epicycle is retrograde, the motion of 

 the planet at 6, as well as that of b B, is in advance, and a + nl repre- 

 sents the whole velocity : consequently, when the epicycle is retro- 

 grade, the pericentral motion is always direct. Observe that we name 

 the deferential motion with respect to the centre, and the epicyclic 

 motion with respect to the centra of the epicycle : as explained in tlie 

 article cited, a motion may be direct with respect to one, and retro- 

 grade with respect to the other. 



We shall now consider the trochoidal mode of viewing the subject, 

 previously to combining the two. Let the circumference of one circle 

 roll upon that of another, any point on, inside, or outside of the rolling 

 circle (if outside, of course supposed to be fixed to it by a carp-ing arm) 

 describes a curve by the motion compounded of the motin of the 

 rolling circle round ite own centre, and the motion of that 

 round the centre of the fixed circle. Three cases may be supposed, as 

 in the following diagrams (fig. 2). The two convexities may be opposed, 

 that is, the rolling circle may roll outside the other ; or the concavity 

 of one may fit the convexity of the other. This last divides into two 

 cases : first, when the rolling circle is the smaller, in which case it ml IK 

 entirely inside the other ; next, when the rolling circle is the larger, in 

 which case the fixed circle is always inside the other. Now each of 

 these cases may easily be reduced to a much more intelligible planetary 

 system, by which much of their difficulty will be removed. It 

 be observed, that when the convexities are opposed, the trochoiil.il 

 system is called e/n'. trochoidal, and when concavity fit* convexity, hypo- 

 trochoidal. We call the radius of the fixed circle r, and of the rolling 

 circle R. 



1. Every epitrochoidal system is a planetary system in which tlm 

 epicycle is direct. Taking the circle which rolls entirely outside the 



