282 MISCELLANEOUS METHODS AND APPLICATIONS. [CHAP. XII. 



7. Two particles, of masses M and nM, are attached respectively to a 

 point of a thread distant a from one end and to that end, and the other end 

 is fixed to a point on a smooth table on which the particles rest, the thread 

 being in two straight pieces containing an obtuse angle TT - a. Prove that, if 

 the particle nM is projected on the table at right angles to the thread, the 

 initial radius of curvature of its path is a (1 +n sin a cos a). 



8. Two particles P, Q of equal mass, are connected by a thread of length 

 I which passes through a small hole in a smooth table ; P being at a distance 

 c from the hole and Q hanging vertically, P is projected on the table at right 

 angles to the thread with velocity v ; prove that the initial radius of curvature 

 of Fs path is 2cv 2 /(v 2 +cg). Prove also that, if Q is projected horizontally 

 with velocity v, the initial radius of curvature of Q's path is 



*250. Initial motions of Rigid bodies and connected 



systems. No new method is required for the solution of problems 

 concerning rigid bodies of the same kind as those considered in 

 Articles 247 and 248 ; the only point to be attended to that did 

 not occur in those Articles is that the system of kinetic reactions for 

 a rigid body reduce to a couple together with the kinetic reaction 

 of a particle of mass equal to that of the body and moving with 

 the centre of inertia. It is however worth while to remark that 

 the expression for the moment of the kinetic reaction of a rigid 

 body about the instantaneous centre at an instant when the 

 velocity vanishes is the product of the angular acceleration and 

 the moment of inertia of the body about an axis through the 

 instantaneous centre perpendicular to the plane of motion [Ex- 

 ample 2 of Article 220], and thus the equation of moments about 

 the instantaneous centre takes, in initial motions from rest, a very 

 simple form. Further it is worth while to notice that the kinetic 

 reactions can always be expressed in terms of a finite number of 

 geometrical quantities which are unconnected by any geometrical 

 equations. This can usually be effected in a simple manner by 

 help of the principles of relative motion laid down in Article 40, 

 and the methods are of the same character as that adopted in 

 Article 247. It may however happen that such methods are 

 difficult of application, and when this is so we may begin by 

 writing down the geometrical relations that hold between the 

 coordinates of the points in any position. If we differentiate these 

 relations twice with respect to the time, and, in the results 

 obtained, substitute for every first differential coefficient of a 



