220 Dynamics. 



therefore the whole integral is 



-_. 



* 



64a* 4a 



Jr 



x 



h \8a 256 a 



To determine the constant Z), we observe that when a? = h, 

 t 1 ' must be equal to 0, and that the arc BM' becomes EM'Z ; in 

 this case, therefore, we have, 



Ba 256 a 



Substituting the value of D obtained from this equation in the 

 expression for % t", making x = h in order to have the entire in- 

 tegral, and observing that BM' becomes then zero, the result will 

 be 



>*"= |J 

 \ 



X + 



h \8a 256 a 2 



But, by taking n for the ratio of the circumference of a circle to 



BM'Z 7i 

 its diameter, we have - = ; accordingly, 



g \8a 256 a 2 



Comparing this value of J' with that of t f found above, we shall 

 have, 



:f;..n |J: IT ('A 

 \5- \ ^ \8a 



256 a 2 



a 256 a 



a quantity in which - is the versed sine of the arc described dur- 

 ing a semioscillation, radius being 1. 



