﻿504 Dr. J. H. Vincent and Mr. 0. W. Jude on 



Frequencies in Simple Ratio hut Unequally Damped. 



The design of the table allows o£ an adjustment in the 

 friction which enables one to damp unequally the motion of 

 the table and the tracing-point. For instance, let the tracing- 

 point and the attaching rods be kept as light as possible, 

 while the table is released from its suspending thread, loaded, 

 and supported on an oiled glass plate by the lower end h 

 (fig. 2) of its axial needle. The damping of the motion of 

 the table can be made as large as is wished with respect to 

 the damping of the tracing-point by adjusting the load. 



If one pair of pendulums is clamped, say those bearing the 

 tracing-point, and the other two set to draw a circle, when 

 the tuning is accurately adjusted, and the friction small as 

 before, the needle-point does not draw a recognizable spiral, 

 but clears the smoke away from an annular surface. If, 

 however, the motion of the table be sensibly damped we get 



The chief feature of this curve is the regularity of the- 

 spacing of the successive branches. The original was 

 measured in a random direction through the centre on a 

 travelling microscope, and it was found that the distance 

 between the branches was, as nearly as could be measured,. 

 •100 cm. No change in the distance could be detected until 

 one approached within a single turn of the point of rest. 

 The friction was in this case almost entirely due to that of 

 the axial needle in contact with the oiled glass plate. This 

 suggests that the trace is the orbit of a point moving with 

 an acceleration towards a fixed centre varying directly 

 as the distance, under a constant frictional resistance. A 

 solution to this dynamical problem is that the orbit may be 

 the involute of a circle of radius /, the acceleration towards 

 the centre being k times the distance from the centre, and 

 the retardation being 2 Ik, On this view, the circle of which 

 the trace is the involute has a radius of *016 cm., so that if 

 we regard the friction as being strictly independent of tho 

 velocity, the tracing-point will come to rest '016 cm. away 

 from the frictionless position of equilibrium. The motion 

 departs from one of uniform angular velocity about the 

 centre of force by a term involving merely the square of the 

 ratio of I to r, so that the tracing-point moves with sensibly 

 uniform angular velocity until it closely approaches the end 

 of its path. 



Fig. 33. If two direct circular motions are compounded,, 

 whose frequencies are in the ratio 1:2, the curve drawn is the- 

 limacon of Pascal, cardioide, or trisectrix, according as their 

 amplitudes are in the ratios 3:1,2:1, or 1:1 respectively,. 



