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S. H. mo

areas are described by the radius-vector from the centre in equal fons in times. Hence the general proposition of § 63.

different directions.




be the Cartesian specification of the first of the given motions;
and so with varied suffixes for the others;

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l, m, n denoting the direction cosines,

half amplitude,


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the proper suffix being attached to each letter to apply it to each case, and w denoting the common relative angular velocity. The resultant motion, specified by x, y, z without suffixes, is


x = Σl,a, cos (wt - e,) = cos wtΣl,a, cos e, + sin wtΣl,a, sin €1,
y = etc.; = etc.;


or, as we may write for brevity,


R = Σn1a, cos ε ̧,



P = Σ l1a, cos ¤ ̧,
Q = Em, a, cos e,,

x = P cos wt + P' sin wt,
y = Q cos wt + Q' sin wt,
z = R cos wt + R′ sin wt,

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¿ = (P2 + Q2 + R2) 3,



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The resultant motion thus specified, in terms of six component simple harmonic motions, may be reduced to two by compounding P, Q, R, and P', Q', R', in the elementary way. Thus if

Š "'

C' = (P12 + Q'2 + R12)3,





P' = Σ l,a, sin e ̧,
Q'Em, a, sin e,,
R' = Σn,a, sin e ̧.



v = 7



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the required motion will be the resultant of (cos wt in the line
(λ, μ, v), and ¿' sin wt in the line (λ', μ', v'). It is therefore mo-
tion in an ellipse, of which 25 and 25' in those directions are



SH. motions in different directions.

conjugate diameters; with radius-vector from centre tracing

equal areas in equal times; and of period

H. motions of different

67. We must next take the case of the composition of simple

kinds and harmonic motions of different periods and in different lines. In

in different lines.

general, whether these lines be in one plane or not, the line of motion returns into itself if the periods are commensurable; and if not, not. This is evident without proof.


If a be the amplitude, e the epoch, and n the angular velocity in the relative circular motion, for a component in a line whose direction cosines are λ, μ, v—and if έ, ŋ, ¿ be the co-ordinates in the resultant motion,

έ=Z.λa, cos (n,te,), n=Σ.μ, a, cos (n,te,),



Z. v,a, cos (n,t — €).

Now it is evident that at time t + T the values of έ, 7, ¿ will recur as soon as n ̧T, n,T, etc., are multiples of 2π, that is, when Tis 27 2π


n2 n2

If there be such a common multiple, the trigonometrical functions may be eliminated, and the equations (or equation, if the motion is in one plane) to the path are algebraic. If not, they are transcendental.

the least common multiple of

68. From the above we see generally that the composition of any number of simple harmonic motions in any directions and of any periods, may be effected by compounding, according to previously explained methods, their resolved parts in each. of any three rectangular directions, and then compounding the final resultants in these directions.

S. II. motions in two

69. By far the most interesting case, and the simplest, is rectangular that of two simple harmonic motions of any periods, whose di



rections must of course be in one plane.

Mechanical methods of obtaining such combinations will be afterwards described, as well as cases of their occurrence in Optics and Acoustics.

We may suppose, for simplicity, the two component motions to take piace in perpendicular directions. Also, as we can only have a re-entering curve when their periods are commensurable, it will be advisable to commence with such a case.

The following figures represent the paths produced by the S. H. mo

tions in two rectangular directions

combination of simple harmonic motions of equal amplitude in two rectangular directions, the periods of the components being as 1:2, and the epochs differing successively by 0, 1, 2, etc., sixteenths of a circumference.

In the case of epochs equal, or differing by a multiple of π, the curve is a portion of a parabola, and is gone over twice in opposite directions by the moving point in each complete period.

For the case figured above,


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x = a cos (2nt — €), y = a cos nt.

x = a {cos 2nt cos e + sin 2nt sin e}

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which for any given value of € is the equation of the correspond-
Thus for € = 0,

ing curve.

1, or y


y2 = = (x + a), the parabola as above.

S. H. motions in two rectangular directions.

Composition of two uniform

circular motions.

For € =







the equation of the 5th and 13th of the above curves.

In general

we have


1. or a2x2 = 4y2 (a2 — y3),



x = a cos (nt + e), y = a cos (n,t + €),

from which t is to be eliminated to find the Cartesian equation of the curve.

70. Another very important case is that of two groups of two simple harmonic motions in one plane, such that the resultant of each group is uniform circular motion.

If their periods are equal, we have a case belonging to those already treated (§ 63), and conclude that the resultant is, in general, motion in an ellipse, equal areas being described in equal times about the centre. As particular cases we may have simple harmonic, or uniform circular, motion. (Compare § 91.) If the circular motions are in the same direction, the resultant is evidently circular motion in the same direction. This is the case of the motion of S in § 58, and requires no further comment, as its amplitude, epoch, etc., are seen at once from the figure.

71. If the periods of the two are very nearly equal, the resultant motion will be at any moment very nearly the circular motion given by the preceding construction. Or we may regard it as rigorously a motion in a circle with a varying radius decreasing from a maximum value, the sum of the radii of the two component motions, to a minimum, their difference, and increasing again, alternately; the direction of the resultant radius oscillating on each side of that of the greater component (as in corresponding case, § 59, above). Hence the angular velocity of the resultant motion is periodically variable. In the case of equal radii, next considered, it is constant.

72. When the radii of the two component motions are equal, we have the very interesting and important case figured below. Here the resultant radius bisects the angle between the component radii. The resultant angular velocity is the arithmetical mean of its components. We will explain in a future section

(§94) how this epitrochoid is traced by the rolling of one circle Composi

tion of two uniform circular motions.

on another. (The particular case above delineated is that of a non-reëntrant curve.)

73. Let the uniform circular motions be in opposite directions; then, if the periods are equal, we may easily see, as before, § 66, that the resultant is in general elliptic motion, including the particular cases of uniform circular, and simple harmonic, motion.

If the periods are very nearly equal, the resultant will be easily found, as in the case of § 59.

74. If the radii of the component motions are equal, we have cases of very great importance in modern physics, one of which is figured below (like the preceding, a non-reëntrant curve).

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