THEORY OF VIBRATIONS 35 



means of three links connecting any three points of it to any 

 three fixed points in the plane*. Similarly any rigid three- 

 dimensional structure can be anchored firmly by six links 

 connecting six points of it with six points fixed relatively 

 to the earth. 



Proceeding to the vibrations of a multiple system about 

 a configuration of equilibrium, we begin as before with the 

 examination of a few particular cases. 



Take first the oscillations of a particle in a smooth bowl of 

 any continuous shape. By means of suitable constraints, the 

 particle may be restricted to oscillate in any given vertical 

 plane through the lowest point 0, e.g. by confining it between 

 two frictionless guides infinitely close to one another. In 

 general there will be a lateral pressure on one or other of these 

 guides, which will however vanish if the plane in question 

 passes through either of the principal directions of curvature 

 at 0. Hence two modes of free simple-harmonic vibration, in 

 perpendicular directions, are possible, with speeds 



nx-Vto/A), n, = ^(g/R,\ (1) 



where B^, R 2 , are the radii of curvature of the principal sections 

 at 0. On account of the assumed smallness of the motion, 

 these vibrations may be superposed. The result is, if x, y be 

 horizontal rectangular coordinates through 0, 



x = A l cos n t t + A 2 sin rz^,] 

 y = Bj cos n^t + B 2 sin nj.) 



Since this contains four arbitrary constants, we can adjust 

 the solution to given initial values of x, y, x, y. 



This case is very neatly illustrated by Blackburn's pen- 

 dulum ( (Fig. 15). A weight hangs by a string CP from a point 

 C of a string A CB whose ends A , B are fixed. The strings being 

 supposed destitute of inertia, the point P will always be in the 

 same plane with A, B, C. Under this condition the locus of 

 P is the ring-shaped surface generated by revolving a circle 



* Provided the directions of the three links be not concurrent (or parallel). 

 There is a proviso of a more complex character in the case which follows ; but 

 such details need not occupy us here. 



t H. Blackburn, Professor of Mathematics at Glasgow 184979. 



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