393 



TROCHOIDAL CURVES. 



TROCHOIDAL CURVES. 



394 





The diagram in MOTION, col. 796, exemplifies the three species of 

 trochoids. 



To suppose the epicycle infinite leads to nothing : but if we now 

 take a trochoidal system, and suppose the rolling circle infinite, we have 

 what is equivalent to a straight line rolling round a circle to which 

 it is always tangent, and carrying a planet at the end of an arm which 

 U supposed fixed to the straight line. If the planet be on the straight 

 line, we have obviously the INVOLUTE of a circle ; for the unrolling 

 thread described in the article cited may be considered as the part of 

 the rolling tangent which has been in contact with the circle since the 

 beginning of the motion. If the planet can pass through the centre 

 of the fixed circle, the curve becomes the spiral of Archimedes. These 

 spirals must be made complete by using both positive and negative 

 values of the radius vector. [SpiUAL.] They are not of much use 

 except to the mathematician. 



The various classes- of trochoidal curves must be studied to aid in 

 forming a proper conception of the effect of combined circular motions : 

 one remarkable instance of their application is that of the apparent 

 motions of the planets. These a]ij,arent motions must be exactly 

 what would take place if the earth were at rest, the sun revolving in 

 an orbit about the earth, which orbit is the deferent ; while the sun 

 itself is the centre of an epicycle equal to that of the planet's orbit, in 

 wliich epicycle the planet moves. And since the epicycle and deferent 

 are convertible, we may in the case of one of the superior planets use 

 the planet's larger orbit as a deferent, and the sun's smaller orbit as an 

 epicycle. Again, since the planet of the smaller orbit always has the 

 greater angular velocity, the ratio of the epicyclic to the deferential 

 velocity, or n, is always, the smaller orbit being the epicycle, greater 

 than unity. Hence, when the planet is retrograde, the loop of its 

 apparent orbit is always turned towards the earth, so that the apparent 

 magnitude of the planet is always greatest in the middle of its retro- 

 gradation. To test these things, we ought to have angles of longitude 

 measured in the plane of the planet's orbit ; but since the inclinations 

 of those planes to the ecliptic are not very great (with the exception of 

 the new planets), angles measured on the ecliptic, or longitudes com- 

 monly so called, will do as well for illustration. 



When the planet begins and ends its retrogradation, the tangent of 

 the apparent orbit passes through the earth, and the planet, moving in 

 that tangent, does not sensibly change its apparent position in the 

 heavens for several days : it in then called ttatitmary. The angle 

 between these two stationary points in the angle of retrogradation, 

 for which a formula has been given. But we shall simply notice the 

 general figure of the apparent orbits, giving the ratio of the radius of 

 the deferent to that of the epicycle, when the smaller of the two 

 orbits (gun's round the earth, and planet's round the nun) is the epi- 

 cycle, and the larger the deferent, and also the ratio of the angular 

 motions. 



a ~ b n a - b n 



Mercury 



Venus 



Mart 



Small ) 

 PUneto ] 



2-58 



1-38 



1-52 



2-4 



2-8 



4-1S 

 1-63 

 1-88 

 3-6 

 to 4-6 



Jupiter 

 Saturn 

 Urauus 



J-20 

 9-34 

 19-18 



11-86 

 2S-78 

 84-01 



All these must belong to looped curves, going round for ever, and 

 never reuniting, in consequence of the practical incommensurability of 

 the values of n. The angles of the descent are for Mercury, 180"-:- 

 (4-15-1) or 67; for Venus, 28t>; for Mars, 205; for the small 

 planets, from 70" to 50 ; for Jupiter, 16" ; for Saturn, 64 -. for 

 Uranus, 2J. The doubles of these angles will be the angular distances 

 between the lower points of two loops. Thus, to construct the orbit 

 of Mi-reury, take a deferent of which the radius is about 24 times that 

 of the epicycle, draw the apocentral and pericentral circles, lay down a 

 succession of radii each at 57 to the preceding, draw the ascents and 

 descents in the manner of the figure in I., and the result will give a 

 sufficient idea of the orbit for mental purposes. The time of the 

 planet moving from the lower part of one loop to that of the next is, 

 roughly, a revolution in the epicycle ; that is, a year of the inferior 

 planet for each inferior planet, and a year of the earth for each 

 superior planet. 



In the case of the moon we have a the radius of the earth's orbit, b 

 that of the moon round the earth, o-=-6 is about 400 and n is about 13 ; 

 whence, since 13 is less than V*00, or 20, the moon's real orbit round 

 the sun has neither loops nor flexures, but is always concave to the 

 sun. [MOON.] 



The mathematical part of this subject labours under the same dis- 

 advantage as the more popular explanation, namely, the adoption of the 

 trochoidal mode of viewing the curves, which, though it gives really 

 the same equations as the planetary mode, embarrasses the question 

 by introducing more complicated formulae. If we take the line drawn 

 from the centre to an apocentre of the curve for the positive fart of 

 the axis of x, and as before, let <f> be the deferential angle, nip the 

 epicylic angle, x and y the rectangular co-ordinates of a point in the 

 curve, r and 9 the polar co-ordinates, a and b the radii of the deferent 

 and epicycle, we have (1) 



x a coa + b cos wf>, y = a aia <p + b B'IU n<j> 

 which are the fundamental equations of the curve. Here a and b are 



ooth positive, <f> (and also 6) vanishes at the apocentre, which is on the 

 positive side of the axis of x, and has its proper sign. But if we 

 place a perieentre on the positive side of the axis of x, other things 

 remaining the same, we have (2) 



x = a cos b cos H(J>, y = a sin <f> - b sin n<j>. 

 If the last terms stand thus, (3) 



x = a cos tp + 6 cos n(p,y = a sin <p + sin n$ 

 they may be reduced to (4) 



+ 4 sin (-$>) 



or the case is one of the preceding, the ratio of the velocities being n 

 and not n. If the first terms be negative, change the signs of x and y, 

 and let the curve thus found make a half revolution : but if the first 

 terms differ in sign, the equation is reduced to one of the preceding 

 cases by changing the sign of <t> (and of n, if necessary), which shows 

 that what we had was the equation to the curve as described by a con- 

 trary motion of the deferent, All this, however, has only reference to 

 the reduction of a particular equation to the form (1), which we shall 

 use throughout. From (1) we readily find 



6cos(a-l)^ . . (5) 

 or the passage from apocentre to perieentre, or from >- = a-t-4 to r=a 

 b, is performed, while cos (n 1) <j> changes from 1 to 1, oF^l <j> 

 from to TT, or to *, according as n is > or < 1. We also have 

 a sin<6 + 4 sin n<t> 



tan 6 = 







a cos + b cos nij> 



To determine the period of retrogradation, if any, we must differentiate 

 the preceding, which gives 



dB 

 ^rf* = ! -t-4' + oi(l+) cog(n-])4> . . .(7) 



and there is retrogradation when the second side is negative, from 

 whence the formula already given is deduced, and the condition that 

 nb must be numerically greater than a (a, being throughout supposed 

 greater than b). The following formula will help to obtain this 

 condition: 



r a' + Vn 1 t_ _ (a-y)(n'6'-q) , g 



lo4(l+n)J = a*P(l+nf 



From the preceding formula (7), it is easy to obtain ItJ'rdB, the 

 area included between two radii and the curve. But this is subject to 

 the interpretation of that part of the area which is contained between 

 two radial tangents to a loop, and the lower part of the loop being 

 negative : the total result of which is that the area inside the loop is 

 counted twice, and all the rest once. 



To find the arc and points of contrary flexure we must deduce the 

 following equations (9, 10) : 



do? dtf- 



dx d-y dy d-x 



d$ dp ~ dj> W = a2 + 6 '" +ab ( nS + n > 08 fa- 1 )* 



The second formula changes sign at the points of contrary flexure, 

 whence.the criterion before given is determined, by help of the follow- 

 ing (11) : 



f oP+Pj* 13 



la4(- + n)J = 





The various double points are thus determined. Let <f> and <f> be 

 the two deferential angles belonging to a double point, the first 

 when tho curve passes through it the first time, and the second 

 for the second. Since the value of r is the same at points, we have 

 cos ( 1) 4>=cos (n 1) <(>', whence we have 



(n -l)f' = 2k* (n - 1) . . (12) 



k being a whole number, positive or negative. If ir-=-(n 1), tho angle 

 of descent, be called p, we have $' = 2 kp +<)>; and if we substitute 

 this in the equations (1) remembering that n/i = ir + ju, we find, by 

 ordinary trigonometrical development, that the following equations 

 exist between the co-ordinates of every double point : 



x = cos 2 leu x + sin 2 fc/t . y, y = sin 2 k n . x + cos 2 k n . y. 



These two equations should be really the same, and they are so 

 if we take the lower signs, in which case they amount to the 

 following : 



y = tan k n . x, 8 = it p, or Jt ^ + IT. 



If the higher signs be examined, it will be found that the two 

 equations are not identical except when 2 k n = 2 / rr, I being a whole 

 number, which can be true only when is a commensurable fraction. 

 But this case being further examined, shows that no double points are 

 indicated, but only that the curve itself is repeated in all its poiiUt 

 after a proper number of revolutions of the deferent, as is otherwise 

 sufficiently obvious. Consequently all the double points of a trochoidal 



