335 



TROCIIOIDAL CURVES. 



TROCHOID.VL CURVES. 



otfier, p U the point describing the curve, and T w has rolled over T V 

 since p was at B. Now o M is constant, being F + R ; so that, since r 



Fig. 5. 



rolls uniformly round M, here is a planetary system in which o M is the 

 radius of the deferent, M p that of the epicycle (which is direct), and 

 the epicyclic velocity is to the deferential as the angle c M P to A o M. 

 Now the arcs T w and T v are equal, whence A o T being <f>, and T M w 

 being if, we have r<(> = Ri|, or 



CMP=( +lj^;so that is the ratio of the velocities. 



Hence, looking at the double generation of every planetary system, 

 we have either 



or a = MP, & = 



R ' 



. 



* T R 



2. Every hypotrochoidal system in which the rolling circle is the 

 larger of the two is a planetary system in which the epicycle is direct. 

 I lore, in the proper diagram, w T has rolled over v T, as before, o M and 

 M P are the radii of the deferent and epicycle, and the epicyclic 

 velocity is to the deferential as the angles CMP and A o M ; and 

 o M = B r. If A o M and T x w be <p and if, we have, from the equal 

 arcs TW and TV, Rtf> = r<f>, and 



Cf \ H _ Y 



1 J <t> ; whence ^- a the ratio of the vclo 



cities. I 



We have then either 



B-F 

 ; 



Ora=HP, i=B-F, = ^j. 



3. Every hypotrochoidal system in which the rolling circle is the 

 smaller is a planetary system in which the epicycle is retrograde. In 

 the proper figure it is now evident enough that o M, the radius of the 

 deferent, is T R; and that AOM and WMT being <f> and ty, we have 

 s^=r<t>, from the equal arcs rvandTW. It is pl.-vin also that M p, 

 the radius of the epicycle, moves with a retrograde velocity. More- 

 over, the epicyclic and deferential velocities are as the angles c H F and 

 A o M, and (R^ being = F<p) 



and 



for the ratio of the velocities. 



ABTS AMD SCI CIV. VOL. VIII. 



Either then 



l = F R, 5= 



MP, 



F R 

 " R ' 



R 

 F R' 



To distinguish the two hypotrochoidal systems, which have very 

 different properties, let that one in which the rolling circle is smaller 

 than the fixed, so that the curve lies entirely inside the fixed circle, be 

 called the internal hypotrochoidal system ; and that in which the rolling 

 circle always contains the fixed circle, and in which the curve is entirely 

 without the fixed circle, the external hypotrochoidal system. It appears 

 then that a planetary system with a direct epicycle belongs to both the 

 epitrochoid and the external hypotrochoid ; while one with a retrograde 

 epicycle belongs to the internal hypotrochoid. 



We now take the converse problem namely, given a planetary sys- 

 tem, to find the corresponding trochoidal systems. This might be 

 easily done algebraically from the preceding results, but a simple geo- 

 metrical construction will much assist the beginner, who rarely can get 

 the true phase of a figure out of formula;. It is required first to con- 

 struct the real velocity and direction of a planet at any point of its 

 curve. The epicycle is, at any given instant, moving forward perpen- 

 dicularly to the radius of the deferent with the velocity o, while the 

 planet is moving perpendicularly to the radius of the epicycle with a 

 velocity n 6. The composition of these two velocities gives the real 

 motion and .direction of motion of the planet for the time being, and 

 shows us how to draw the tangent of its curve. 



Fig-. 3. 



Let o M and M r (fig. 3) be radii of the deferent and epicycle (not 

 drawn) ; and from p, the planet, draw p x and p T perpendicular to o M 

 and M p. Make pxtopvasotonft, and complete the parallelogram 

 p x Y z. Then p x represents the motion, for the instant, of the whole 

 epicycle, and p Y the motion of the planet in the epicycle ; whence p z 

 represents the planet's actual velocity, and p z is tangent to its curve. 

 In the first of the figures the epicyclic motion is direct, and in the 

 second retrograde, the arrows showing the direct motion of revolution. 

 Also, for variety, the planet is placed much nearer to its pericentre in 

 the second figure than in the first. 



Draw p R 8 perpendicular to r z, meeting o Q and o M in s and it. 

 Then, the sides of the triangles Q P s, P M n, being severally perpendi- 



c o 



