Ml 



TROCHOIDAL CUUVEB. 



riK..-IIi'11'AI. CTIIVES. 



are *till positive throughout this Motion, and the curve is a series u{ 

 shorter concave descents. The final ellipse hat a + 4 and a 6 for the 

 etui-axes. 



Both the trochoidal systems are internal hypo-systems, the planet 

 being inside the larger of the rolling circles, and outside the smaller. 

 At the extreme case, the ellipse, the two systems are the same, anil in 

 both i-ttrf the fixed circle becomes double of the rolling circle in 

 diameter. Before the extreme case, the greater rolling circle belongs 

 to the greater fixed circle, and the less to the less. 



V. When n incrratet ntgatirtly from 1 to a : b. Something 

 resembling the last continues (the angle of descent still diminishing) 

 until that angle is too small to allow of a descent wholly concave. By 

 the time n becomes V (-r- &) the descent is almost a straight line, 

 except very near the apocentre, and when n is greater than <J (a-r-b), 

 taken negatively, there are points of contrary flexure dividing the 

 convex from the concave part as before. 



Then points of contrary flexure do not, as before, return to the peri, 

 centre at which they begin; but as n varies from V (a-f-i) to 

 o-t-6, they run up the arc of the curve, and unite at the npocentre, 

 when = 6-f-o, in a cusp. This cuspidated curve is the hypocycloid, 

 and up to this point both apoccntral and pericentral motions are 

 direct. 



The only change that has taken place during this time in the tro- 

 choidal systems is a continual increase of both circles in one of them, 

 and a continual diminution of both in the other. At the hypocycloid 

 both the rolling circles are thus brought to pass through the planet, 

 and the fixed circles become equal. The common radius of the fixed 

 circles U a + 6, those of the roiling circles are a and b. 



VI. H'hrn n incrtatti ntgalirely from a : b. We left off with the 

 hypocycloid at which the apocentrnl motion (in revolution) is nothing. 

 This apocentral motion afterwards becomes retrograde, the pericentral 

 continuing direct, so that apocentral loops begin to be formed ; and the 

 central angle of retrogradation, in which the planet ptsam through the 

 higher part of a loop, may be found from a preceding formula. 



As ti increases, these loop* become nearer and nearer, and at last 

 begin to interlace, the angle of descent continually diminishing. In 

 the trochoidal systems, both circles of one of the bypo-systcms in 



without limit with n, while in the other the fixed circle |<er- 

 tually becomes more nearly equal to the deferent, while the rolling 



circle diminished without limit As it increases, each of the loops 

 nore and more nearly coincides with the epicycle, and the c-xti. ii-- 

 imit is a circle of no radiut, carrying the planet on an arm equal to 

 .he radius of the epicycle, and revolving on a circle equal t<i tin- 

 ieferent : or a revolution in an epicycle whose centre is fixed. Tliis 

 imit of course is never attained ; or perhaps the analyst would say, 

 ihat when n is infinite, the planet goes over the whole space between 

 the apocentral and pericentral circles. 



When the deferent aud epicycle are equal, or a = b, the epicycloid 

 Becomes the apocentral circle, aud the hypocycloid the straight line. 

 The other curves take various extreme forms, as in the foil 

 diagrams : 



There is never any sensible arc of retrogradation ; all the retrograda- 

 tion, as it were, taking place suddenly at the pericentre, that is, at the 

 centre. We shall leave the further investigation of these cases to the 

 student. 



When n = 2, the angle of descent is always 180, and the epicycl.id 

 bag the rolling and fixed trochoidal circles equal, and is a curve shaped 

 somewhat like a heart, whence it is called the cardioide. 



On these curves generally we may remark, tliat their minor modi- 

 fications of form are very various. The descent through a point ,.f 

 contrary flexure, for example, gives two curves of very different figures, 

 ng as the angle of descent is more or less than a right angle ; 

 and the loops may be made very small or very large, compared with 

 the rest of the curve. It must be particularly noticed also, that l.y 

 the angle of descent we mean simply the angle between the a po. 

 and pericentral distances, not the vectorial angle described in p.in^ 

 from one to the other : the latter may become greater than this angle 

 of descent before the pericentre is arrived at, and then diminish down 

 to the angle of descent. 



Let us now suppose the deferent to become a straight line, or a = <x> , 

 and let the centre of the epicycle be carried along this line with .-i 

 velocity a, while the planet moves in the epicycle with an angular 

 velocity r in theoretical units. Curves will now be described which 

 are called trochoidt, among which the CTCLOID has the same place as 

 the epicycloid among the epitrochoids, or the hypocycloid among the 

 hypotrochoids. The forms can be readily imagined by conceiving the 

 deferent to be a circle of great radius. It is now of no consequence 

 whether the epicyclic motion be direct or retrograde. If r be very 

 great, we have a series of interlacing loops ; when becomes less, these 

 loops separate; when a rb, the loops degenerate into cusps, and we 

 have the cycloid ; and when a is greater than rb we have only a series 

 of undulations with points of contrary flexure. All this may be i 

 Rented by a trochoidal system in which a circle rolls upon a straight 

 line, carrying with it an arm to which the planet is attached ; and tin- 

 trochoid is looped, cycloidal, or wavy, according as the planet is with- 

 out, on, or within, the rolling circle. It will be a good exercise for 

 the student to deduce the centre and radius of the rolling circi. 

 a construction similar to that hereinbefore employed for deducing 

 the elements of the trochoidal from those of the planetary system.' 



