M 



TROCHOIDAL CURVES. 



TUOCHOIDAL CORV1-X 



cular to those of P * x, these time triangle* are similar to one another. 



I'-:. 



pq ( = a) : QS : : px : XI : : a : &, or qs i 



a 

 FM (=6) :MB::ix:xp::*o:a,orMB--. 



In the second figure, or when the epicycle U retrograde, it will 

 always be found that K and a are in o M and o q produced ; but the 

 first figure, as drawn, is good only for the case in which n is greater 

 than 1, as was supposed in the construction, and is seen in the result, 

 since q s (or n 6) is greater than q o (or b), and B u (or a : N) is less than 

 o M (or a). If H had been less than 1, t s would have passed through 

 o g ; but in all oases oi direct epicycle, one of the sides o u and it q is 

 cut externally and the other internally ; while in every ease of retro- 

 grade epicycle, both are cut externally. We have then 



qs = In 



ros-i(i + ), 



Epicycle ratrognds) / 1 \ 



loH-a(l +-j, 



Epicycle direct 



-40- 



--!, qs 



n 



In 



a 



' n 



We can now immediately show that every planetary curve can be 

 trochoidally described in two ways, it being already known that every 

 trochoidal curve can be planetarily described in two ways. It appears 

 that o B and o s are the same for every position of the planet in a given 

 system, since they depend only on a, b, and n. Consequently, while 

 the planet moves, B and B revolve in circles about o ; and if we now 

 pass to a trochoidal system in which o B is the radius of the fixed, and 

 M B of the revolving circle, the point r, connected with the revolving 

 circle by the arm H P, will describe a trochoidal curve which is identi- 

 cal with the planetary curve, from the elements of which its circles were 

 obtained. Or o s may be made the radius of the fixed circle, and Q s 

 of the revolving circle, the arm of connection being q p. For instance, 

 U we take the retrograde epicycle, and make 



n 

 , B = frn; theni=P-B = ~> 



which are the equations, already found, of connection between the 

 internal hypotrochoidal system and its corresponding planetary one ; 

 and similarly for the other cases. And the result is, as appears from 

 the figure, that every direct-epicycle planetary system is both epitro- 

 choidal and externally hypotrochoidal, while every retrograde-epicycle 

 planetary system is in two different ways internally trochoidal. We 

 are thus enabled to refer the description of all the trochoidal curves to 

 their corresponding planetary systems, which are much more easily 

 followed, especially when, as is always in 6ur power, we make the 

 radius of the epicycle not exceeding that of the deferent. It has ap- 

 peared that when a = t , the planet has no motion of revolution, at 

 the apocentre in a retrograde epicycle, and at the pericentre in a direct 

 '.a ; the motion must then at those epochs be all from or to the 

 centre, giving curves with such coups as are shown in a former diagram. 

 Mow, looking at the corresponding trochoidal systems, wo see that when 

 a = n 6, qg = qr, and 11 B = >i p, or the point which describes -the 

 trochoid is on the circumference of the rolling circle. In this case the 

 epitrochoid is called an epicycloid, and the hypptrochoid an hypo- 

 cycloiiL And since in all cases the line which joins P with the point 

 of contact of the circles (R or 8) is normal to the curve, or perpendicu- 

 lar to the tangent, it follows that in the epicycloid and hypocycloid, 

 the two chords which join the point that traces out the curve with the 

 two extremities of the central diameter of the rolling circle are, one 

 tangent, and the other normal, to the curve. 



We shall now pas* on to the consideration of the varieties which 

 planetary curves offer ; and first we have to separate some extreme 

 or critical cases from the rest. These are when n = 1, 0, or 1, for 

 we shall now begin to distinguish direct from retrograde epicyclic 

 motion by making n negative in the latter case. When n = 1, p (.'"';/ 1 ) 

 will be always in the continuation of o u, either at E or e, as it was first 

 placed ; consequently the planetary curve is here only the apocentral 

 or pcricentral circle of other cases. If n = 0, p will always coincide 

 with c ; now c describes a circle equal to the deferent, but having its 

 centre at K, o K being equal to the radius of the epicycle. If n = - 1, 

 the angle CMP (Jig. 4) is always equal to M o H, and the triangle* ton, 

 c M r, are similar ; whence K r, or twice I, p, U always a given proportion 

 of BK, the ordmate of a circle: it follows, then, that the planetary 

 curve is an ellipse when n = 1. But when o - 6, the point p ID 

 always at H, in the line o n, and the planetary curve is as much of the 

 straight line OHM extends from twice o M on one side to twice o M on 

 the other. Looking at the trochoiiUl character of these varieties, we 

 have, when n 1, either r = 0, n -= A, or r = 0, B = a ; that is, the 

 curve tlu.'n described is made by a circle revolving round a point in it - 

 circumference ; in both oases we have a circle. But when H - 0, we 

 have r = b, B = 0, or the trochoidal curve belongs to a point connected 



with a circle of no radius, which revolves on a circle of the r.idiim ', ..r 

 O K. This is one of those extreme eases which are rather interpreted 



Fi. 4. 



than perceived [IVTKRPBETATION] : if a circle of no radius revolve 

 from K, it con never make any progress, and the arm, K c, which carries 

 the moving point, is always describing a circle. It is the extreme case 

 of the following supposition : Let a circle of extremely small radius 

 revolve on the circle of radius O K, carrying with it the arm K i 1 ; it will 

 make but little progress on the larger circle in many revolutions, 

 during which c will describe many nearly circular folds very close to 

 each other. Lastly, when n = 1 retrograde, we have r = 2A, B = b, or 

 tho fixed circle has twice the radius of the rolling circle. When a = 4, 

 the point which describes the curve is on the circumference of the 

 rolling circle ; and thus we have the following theorem : When a 

 circle rolls inside another of double its diameter, every point attached 

 to that circle, internally or externally, describes an ellipse ; but every 

 point on the circumference of the rolling circle describes a straight line, 

 the extreme limit of an ellipse. 



Since we can always suppose the lesser of the two circles to be the 

 epicycle, and the greater the deferent, when there is a lesser and a 

 greater, there is yet another extreme case hi which the epicycle and 

 deferent ore equal, so that all the pericontres are in the centre o itself, 

 for now the epicycle always passes through that centre. This case will 

 be best considered after the several other cases of which it is the 

 extreme. We now go on to the general question/ namely , ha\ i > 

 epicycle lees than the deferent, and having both radii given, required 

 the forms of all the varieties of the planetary curve which arise from 

 giving different values to n. 



We may recapitulate the formulas first given with their algebraical 

 character, as follows : The radius of the deferent is o, that of the epi- 

 cycle 6, the ratio of the angular velocity of the planet in the epic} 

 that of the epicycle round the deferent is n, which is negative wli 

 epicycle is retrograde. The angle moved through by the centre of the 

 epicycle since the last epoch when the planet was at its apocentre being 

 if>, that moved through by the planet round the epicycle in the same 

 time is n<f>, and when the planet has come to its pericentre, 4> is 

 180 -f- (11 1), this meaning, when positive, that the radius of the 

 epicycle has gained 180" in direction upon that of the deferent ; and, 

 when negative, that the radius of the deferent has gained 180 upon 

 that of the epicycle. And the apocentral velocity of the planet is 

 a -t- H !>, n having its proper sign, its absolute revolution round the 

 centre being direct or retrograde, according as a + 6 is positive T 

 negative. The pericentral velocity is o n 6, to be interpreted in the 

 same manner. In the corresponding trochoidal systems, r and it being 

 the radii of the fixed and rolling circles, and c the length of the arm 

 which carries the moving point, measured from the centre of- the rolling 

 circle to which it is attached, we have either 



(1\ o 



1--J, a = ^, e = b, 



where n is to have its proper sign, and 



P positive, R positive, denotes an epitrochoidal system ; 



P negative, R positive .... external hypotrochoidal system 



P positive, H negative .... internal hypotrochoidal system : 



or, in fact, that circle only which sees the concavity of the other, is to 

 have its radius counted as negative. This is an induction from tho 

 various previous coles, such as the student of algebra will rv.-idily 



P;: :k' . 



I. When n iliminMa from a great value down to a -=- l>. I 

 first suppose n very great and positive, so that the angle from apoccutro 



to pcrioontre is small, tho apocentral revolution direct, tli> 



ide. Passing to the trochoidal systems, both circle* urc large in 



