26 UNIVERSITY OF MISSOURI STUDIES 



The equation of the curve in figure 8 is : 



(II) y = c(l — cos2nmt); 



that is, while t changes from zero to 1 , v changes from 



n 

 zero through c, 2c, and again c, back to zero. We now sub- 

 stitute Cx for y : 



c (1 — cos 2irnt) = Cx, consequently: 



(III) cos 2Trnt = 1 — — x 



c 



This formula permits us to calculate t, that is, the exact 

 time when any point of the partition is jerked down. But 

 it holds good only for the time from A to B, that is, while the 

 stirrup moves in one direction. As soon as the stirrup reverses 

 its movement a new formula has to be applied, since the move- 

 ment of the partition is of a kind which is mathematically 

 called a discontinuous function. The moment when the stir- 

 rup reverses its movement and the farthest point of the par- 

 tition has been jerked down, the function jumps, so to speak, 

 from this point to the beginning of the partition and the first 

 point, nearest the windows, is jerked up. The formula to be 

 used from B to C is to be derived by substituting (2c — y) 

 fory in (I), since x would now be proportional to (2c — y). 

 We then have the following new equations : 

 (IV) 2c — y = Cx. 



(II) y = c (1 — cos 2irnt), consequently : 



r 



i \ i cos 2-rrnt = x — i . 



c 



This frrmula is valid from B to C, that is for values of 



t varying from _1_ to — • while the partition is being 



2n >i 



jerked upwards. We notice that the only difference between 



the right side of (III) and the right side of (V) is the sign. 



For the same x we obtain the same absolute value of cos 2irnt, 



but in the one case it is positive, in the other negative. Now, 



it is easy to see what this means for the time interval between 



a downward and an upward jerk of any point of the partition. 



