ON THE TA UTOCHRONE IN A RESISTING MEDIUM. 95 



. pi dz 

 andV2<^=J o ^ 



Therefore for tautochronism 



But Fz = ( ~ (1 - fa) dz, 

 J ds * 



therefore -r z = az -j- (1 fa), 



^ r, ^ dx I e~ ki 

 therefore k -j- = a ^- , 

 ds e k8 



the equation of the tautochronous curve. 

 We shall have, if t = when s = s v 



z^ cos 

 ds* e k -l 



must become, when s is expressed in z, 



d*z 

 W ~~*9*> 



a result easily verified, for 



(1), 



The coefficient of dz in (a) is of course that of d 2 z in 

 which shews that if the equation of motion were 



d z s ds 7^_ e*-l 

 dt^ h dr k d?- '" ff ~k~> 



or E = hv + A;v 8 



