MECHANICS 711 This equation, together with the two equations of the curve, is sufficient to determine the motion completely. Also R! and N" being considered positive when acting towards the centre of absolute curvature ; this equation determines R a . Now R 2 is the reaction which prevents P s withdrawing the particle from the osculating plane ; and therefore R 2 =-P (3). (2) and (3) give the resolved parts of the pressure on the curve. with the osculating plane. If the result of the investigation should show that at any time R could vanish, the particle must be treated as free until the equations of its free motion show that it is again in contact with the curve. article A particle moves, under given forces, on a given smooth surface; a to determine the motion, and the pressure on the surface. be the equation of the surface, R the reaction, acting in the normal to the surface, which is the only effect of the constraint. Then, if , /j., v be its direction cosines, we know that A.= rfF 2 dx (2), with similar expressions for p. and v, the differential coefficients being partial. If X, Y, Z be the applied forces on unit of mass, our equations of motion are. evidently, z = Z + E (3). Multiplying equations (3) respectively by x, y, z, and adding, we obtain , . , (4). R disappears from this equation, for its coefficient is Ax + yuy -t- v z and vanishes, because the line whose direction cosines are propor tional to x, &c., being the tangent to the path, is perpendicular to the normal to the surface. If we suppose X, Y, Z to be a conservative system of forces, the integral of (4) will be of the form 1 2 W//y /)/cAj-f^ ( f and the velocity at any point will depend only on the initial cir cumstances of projection, and not on the form of the path pursued. To find R, resolve along the normal, then which gives the reaction of the surface ; p being the radius of cur vature of the normal section of the surface through the tangent to the path, and the mass of the particle being taken as unity. To find the curve ivhich the particle describes on the surface. For this purpose we must eliminate R from equations (3). B this process we obtain two equations, between which if t be eliminated, the result is th differential equation of a second surface intersecting the first in the curve described. If there be no applied forces, or if the component of the ap plied force in the tangent plane coincide with the direction o motion of the particle, then the osculating plane of the path of th particle, which contains the resultant of R and the applied force, will be a normal plane, and therefore the path will be a geodesic on the surface. Thus a particle under no forces on a smooth (or rough) surface will describe a geodesic. Conical 154. An excellent and important ex- pendu- am ple is furnished by the simple pendulum, um when its vibrations are not confined to one vertical plane. When the bob moves in a horizontal plane, the arrangement is called a "conical" pendulum, and it is a very simple matter, a follows, to find the motion. For the vertical componen mg >f the tension of the string must support the weight of he bob ; i.e., vhere a is the inclination of the string to the vertical. llso the horizontal component of the tension must supply lie force wV 2 /R ( 49) requisite for the production of -he curvature of the path, i.e., V 2 Tsinct = ?;i-r-^ . Zsina Eliminating m/T from these equations, we have cosa gl siu 2 a V 2 But, if T be the time of revolution of the bob, Hence gl /I cos a r Zir f > V a or i.e., the conical pendulum revolves in the period of the small vibrations of a simple pendulum whose length is the vertical component of that of the conical pendulum ( 134). To carry the investigation to cases in which the pendulum de scribes a tortuous cjirve, we require (except for approximate results) the use of elliptic functions. We thus obtain, among others, the following results : The motion will be comprised between two horizontal circles. Let the depths of these circles below the centre be b + c and b-c; then the vertical motion of the bob of the pendulum will be the same as that of a point on a simple pendulum of length l z /c per forming complete revolutions in the same periodic time as the spherical pendulum. But for one of the most important applications the deflexion from the vertical is always very small, and it is easy to obtain a sufficiently accurate working approximation without the use of elliptic functions. If we put p and q for the semidiameters of the small elliptic orbit which will then be described by the pen dulum bob, we find for the apsidal angle )- Hence, when a pendulum is slightly disturbed in any way, the motion is to a first approximation elliptic as in 50. But the second approximation shows that this ellipse rotates in its own plane, and in the same sense as that in which it is described, with an angular velocity proportional to its area. Hence the necessity for extreme care, in making Foucault s experiment (presently to be described), lest the path should even slightly deviate from a vertical plane. 155. Another very important and useful example is B!acld>un furnished by Blackburn s pendulum, which is simply a pemlu- pellet supported by three threads or fine wires knotted um * together at one point C (fig. 50). The two other ends of two of them are attached to fixed points A and B, and the third supports the pellet P. The motion of P is virtually executed on a smooth surface, whose A principal curvatures near the lowest point are 1/CP in the plane of the three threads, and 1/PE in the plane perpendicular to them, E being the intersection of the vertical through C with the line AB. Hence for small disturbances of this system, P has a *P simple harmonic motion in the plane __ of the paper whose period is IK JCP/g, and another at The amplitudes right angles to it, with period 2w of these motions are arbitrary, and, with the difference of phase, depend entirely on the initial disturbance. _ Thus we have a very simple mechanical means of producing the combinations treated in 63; for we have only to make PE:PC: :o> 2 :a/ 2 , and give the bob its proper initial motion. 156. When CE is very small compared with CP, we
have a realization of the case of 61, in which the