248 Wisconsin Academy of Sciences, Arts, and Letters. 



when compounded together, to a rectilinear vibration. The peri- 

 odic time of this plane vibration is equal to that of the circular 

 vibrations, its amplitude is double, and its direction is in the line 

 joining the points at which two particles describing the circular 

 vibrations, in opposite directions round the same circle, would 

 meet." 



The theorem maj be illustrated as follows: 

 If, in any space like that represented in Fig. 10, we have a great 

 ^^' ^^' number of spi7is, more or less 



completely filling the space en- 

 closed by the larger circle, and 

 about axes perpendicular to the 

 plane of the paper, the resultant 

 will be equivalent to a spin of 

 definite magnitude about some 

 single axis likewise perpendic- 

 ular^to the to the plane of the 

 paper; the magnitude of this 

 resultant spin being determined 

 by the intensity, relative dis- 

 tances, and number, of the component spins which go to make it 

 up. Eegarding this resultant spin only, the velocity of a particle 

 at any distance from the axis can be decomposed into component 

 velocities, as in Fig. 11, where ~ 



the uniform circular motion 

 of F, from X to Y, can be de- 

 composed into ?=r. cos d and 

 ^=r. sin d, in such a man- 

 ner that the motion of D, to 

 and fro on the line X, and 

 the motion of E to and fro on 

 on the line Y, correspond 

 constantly in position to the 

 motion of F around the cir- 

 cle. In such a case, we say 

 that the circidar harmonic 



motion of F is compounded of two rectilinear harmonic motions 

 along X and Y, of equal periods and amplitude, but differing bj 



