MOTION OF AN OSCILLATING ROD. 163 



will be a fractional quantity. Let it be denoted by e, so 

 that 



^=--(i + e). 

 Hence, the four roots of the biquadrate equation are 



V^\/-(i-e), V^Y^i+e), 



and are all imaginary. Writing p for \/ — (i— e) y and 



<? for \f — ( x + e )j we obtain as the general solution of 

 equation (22), 



or if we substitute angular functions for the impossible 

 exponentials, 



a? = c x (cos pt + \/ — 1 sin pt) +c z (cos pt— V — 1 smpt) 



+ c i (cos qt + y/~i &m qt)+c 4 (cos gt— \/^ism qt). 



This may be more briefly written 



a?=¥ cos pt -f Q, cos qt + R sin^ + S sin g£. 



P, Q, It, S may be determined by the initial conditions of 

 the motion. When t=o, x=o. Hence we have 



o=P+Q (23) 



Differentiate, put t=o, and V for the initial velocity, 



Y=Rp + Sq (24) 



Differentiating twice, and substituting initial values, we 

 have 



0=~Pp*-Q<f; (25) 



