42] CYCLOID AS TAUTOCHRONE. 145 



we may integrate, obtaining 



log y' (z) = 2 - log z + const. 

 Taking the antilogarithm, 



r-N . f x C dS 



5) ^0) = -= = r> 



]/2 d# 

 since s = <p (#). Integrating again, 



6) s = 2c/*.+ d, 



where c and 6? are arbitrary constants. This is the equation of the 

 curve. In order to recognize its ; nature let us square equation 5) 

 writing 



where a = is an arbitrary constant. Solving for ~- 



ox 



If we put ^ = a (1 cos -9*) this becomes 



dz . _ (Z'O' T A cos ^ sin # 



9 ) 



from which 



a (1 -f cos -9 1 ) d& = dx. 

 Integrating, 



x = a (ft + sin %) -f- const., 



or, if x and z vanish together, 



10) 



z - = a (I cos &) . 



These are the equations of a cycloid, 29 p. 83, accordingly the cycloid 

 is not only a brachistochrone for gravity, but also a tautochrone. 

 For a particular cycloid the time of descent is by 2) and 7) 



o 



or putting z = 2^11, 



i 



12) T^=y a 



' 



Putting u = sin 2 #, we easily obtain 



13) T = 



WEBSTEK, Dynamics. 10 



