TROCHOIDAL CURVES. 



TROCHOIDAL CURVES. 



390 



the external hypo-system; and the fixed circle nearly equal to the 

 deferent, and the rolling circle small, in the epi-system ; also the 

 planet is far within the circumference of the rolling circle in the exter- 

 nal hypo-system, and far without it in the epi-system. All this must 

 be collected from the formula:. The curve is then of the foregoing 

 form. 



The angle during which the revolution of the planet is absolutely 

 retrograde is always thus found : Determine <p l from the equation 



which can always be done if the second side be less than unity. Then 

 the retrogradation begins when ^>=;^j, and ends when. 



360 

 f^^l -<?i- 



AM n diminishes, the angle from apocentre to pericentre. increases ; the 

 radii of both circles in the external hypotrochoidal system diminish ; in 

 the epitrochoidal system the fixed circle grows less, and the revolving 

 one greater, both the circumferences approaching the planet. When n 

 grows small enough, being still so large that a < nb, the loops cease to 



interlace, and become separately visible. This goes on until n is so far 

 reduced in value that a = nb, when the perioentral velocity vanishes, 

 both the trochoidal systems have the planet upon the rolling circle, the 

 loops degenerate into cusps, and we have epicycloids, as follows ; 



Towards the end of this division the character of the curve may be 

 much affected by the relative value of 6 and a. At the epicycloid, 

 when = a-r- b, the angle from apocentre to pericentre, as subtended 

 at the centre, is 180 ^- (i - 1) or 180 6 -=- (o 6) in degrees. The 

 nearer a -i- b is to unity the greater does this angle become, and if a 

 be near enough to b, it may be increased to any amount : so that a 

 planet might descend from apocentre to pericentre through thousands 

 of thousands of revolutions before it formed its loop or cusp, and 

 began to ascend, as in the following figures : 



If a be actually equal to 4, all these curves have the lower points of 

 their loops in the centre itself, but as n diminishes down to unity, the 

 descent becomes slower and slower; and when n= a 4- b, or 1, ceases 

 altogether, the apooeutral circle itself being the hypocycloid of this 

 extreme case. 



II. When n diminithei from a-i-b down to 1. \Vhennhasbecomca 

 little less than a-^6, the angle from apocentre to peiicento-e has in- 

 armed, the velocities at both places are direct, or the planet is never 

 retrograde. In the trochoidal systems, it is now without the rolling 

 circle of the external hypo-system, but within that of the epi-system. 



The cusps have given way to points of contrary flexure, as in the 

 following figure : 



The way to find these points of contrary flexure ia as follows : Find 

 . from the equation ; 



then there is one point of contrary flexure when $ = $?', and another 

 when 



-&* 



As n diminishes, these points of flexure, after approaching somewhat 

 towards the apocentre, cease that approach, and begin to return to- 

 wards the pericentre, in which they are lost when n has come down to 

 the square root of a-t-b. But this time they do not unite in the cusp 

 from which they came, but the convex part of the curve disappears, in 

 such a manner as to give a remarkable straightness to the parts adjoin- 

 ing the pericentres. 



When n is less than \/(a-~6) the curve ia always convex, and the 

 angle from apocentre to pericentre perpetually increasing, the descent 

 may be made as slow as we please, as in the following : 



At the limit of this case, when n becomes 1, all descent ceases, and the 

 apocentral or pericentral circle is all that is left, as before described. 



At the beginning of this section of curves, the planet was upon the 

 rolling circle in both trochoidal systems : it immediately passes out- 

 side the rolling circle in the external hypo-system, and inside the 

 rolling circle in the epi-system. Moreover, in the former the fixed 

 circle gets smaller, and the rolling circle approaches the epicycle, so 

 that when = 1, the external hypo-system is merely the epicycle re- 

 volving round a point in its circumference. But in the latter, as n 

 approaches to 1, the rolling circle approaches the deferent in size, while 

 the fixed circle becomes smaller and smaller : its extreme case coin- 

 cides with the extreme case of the former system. 



III. When n diminis/iei from 1 to 0. The angle from apocentre to 

 pericentre, or the angle of descent, as we may call it, which left off 

 infinite at the end of the last section, immediately becomes very great 

 and negative when n is a little less than 1 ; indicating that the descent 

 is performed by the radius of the deferent gaining upon that of the 

 epicycle, instead of the contrary. The curve is always concave to the 

 centre, and at first, in its long folds, much resembles that of the last 

 diagram. But as n diminishes towards 0, the angle of descent becomes 

 less and less, and is only 180 when ?t=0 ; and during this time, the 

 descent becomes more and more circular, until, when n = 0, we have 

 nothing but the circle with centre K and radius K B, as before explained 

 (J*3' !) -^ 8 to the trochoidal systems, the external hypo-system and 

 the epi-system have changed formula) : in the latter, the fixed radius 

 changes from to 6, and the rolling radius from 6 to 0, the planet 

 being always outside the rolling circle : the extreme case has already 

 been explained. But in the former system both radii increase without 

 limit, and become infinite when ?i = 0, the planet being always within 

 the rolling circle. The extreme case must be thus explained [INFI- 

 NITE] : to construct a planetary curve in which the epicyclic motion is 

 very slow compared with the deferential, by means of its external 

 hypo-trochoidal system, we must take both radii, fixed and rolling, 

 very large, and the larger the slower the relative epicyclic motion, 

 without limit. 



IV. When n increase! negatively from to 1. The angle of descent, 

 which left off at 180, now diminishes in amount, being still nega- 



e; and by the time = -1, it brings us, as wo have seen, to an 

 ellipse, which gives - 90. The apocentral and pericentral velocities 



