404 PROCEEDINGS OF THE AMERICAN ACADEMY. 



damped. Consequently the r. m. s. value of an oscillating quantity 

 Vq€-^^ sin wt during any number of complete periods, is either the 

 r. m. s. value of its initial undamped vector multiplied by 



( sin ^ i/ -' ) 



or the r. m. s. value of its mid-interval damped vector, multiplied by 

 sin ^ y '■ — 7 — ^. After Ux passes the numeric 2, the first of these 



rapidly converges to ■ , so that when the oscillatory damping is 



rapid, the r. m. s. value of the oscillatory quantity varies inversely as 

 the square root of the interval during which the summation is effected. 

 If the summation be confined to a single period, starting with radius 

 vector Vq, m=-\ and 



(sm*|/^). Ph.Q. (67) 



' 





V2 



In the case considered, the current is ?' = 8.165 2~iooo< sin 1224.75^. 

 If we take the first complete period oi T= 0.00513 second, with 



sm 



(^ =0.7746; and (sin </> yLA^!! J ^ 0.7746 Vyo^^^-^^^; 



so that the r. m. s. value of the current during the first complete 

 damped oscillation would be 1.396 amperes. 



If the oscillating quantity is a cosinusoid of the t)rpe 



v= V,e-'^coswt. Ph.Q. (68) 



The integral to infinite time of its square: 



/: 



\^dt = ^' (1 + cos' ct>). (Ph. Q.)'^ (69) 



