Page 248

248     Wisconsin Academy of Sciences, Arts, ard 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 may be illustrated as follows:
    If, in any space like that represented in Fig. 10, we have a great
               Fig. 10.            number of spins, more or less
                                   completely filling the space en-
                       Oz4>  By  a<\  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
               cm t  J   /      resultant spin being determined
                                   by the intensity, relative dis-
 tances, and number, of the component spins which go to make its
 up, Regarding this resultant spin only, the velocity of a particle
 at any distance from the axis can be decomposed into component-
 velocities, asin Fig. 11, where      Fig. 11
 the uniform circular motion           y
 of F, from X to Y. can be de-
 composed into =_r. cos 0 and  /      E
 v=r. sin 6, in such a man-
 ner that the Lotion of D, to                     Irv
 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 circular harmonic
 motion of F is compounded of two rectilinear harmonic motions
along X and Y, of equal periods and amplitude, bat differing by