708 PROCEEDINGS OP THE AMERICAN ACADEMY, 



and it would end at the time 



t= T> +T 2 = Tj=T, 



so that one of the turning points of the minor vibration coincides with 

 that of the major vibration, as before. This is true in all cases. 



The complete motion of a point in the second kth. of the string is, there- 

 fore, (a) two ascents, the first during the interval 



o<«r(»-j'i. 



with a velocity 



' T I ' 

 and the second during the interval 



, AHl — x AHfx 1\ AHh-\ 



[ 2 --%\ <t <T X 



,k I J ^ ^ I 

 with a velocity 



AH l-x AH (2_x\ _ AHk-2 . 



followed by (b) a number of descents, the first during an interval centred 

 at 2T/k, with a velocity 



,_AHx A_H(x \\ _ AHl 



g ^ - r i~ t \i~ kJ~~YV 



the second during an interval centred at 3 T/k, with a velocity 

 ,. AHx AH(2 x\ AH2 



9 J rf*' = ^ r - l + -Y\ Kk - 1 )=-?y 



the third during an interval centred at A T/k with the velocity AH/Tk 

 as before, and so on. Iu no case will the curve y = <£ (t) for any point 

 not in the first or last £th of the string have any horizontal portions what- 

 ever, but both the ascending and the descending portions will be some- 

 what "crumpled." 



These facts are very prettily illustrated by the appearance under a 

 microscope of the bright lines upon which the observations of the last 

 section were made. When the observed point is in one of the end £ths 



