ON THE TAUTOCHRONE IN A RESISTING 

 MEDIUM*. 



OUR object is to reduce the problem of the tautochronous 

 curve, when the resistance is equal to Jcv* or to hv + kv z , to the 

 cases in which it is a cycloid, viz. when the resistance is equal 

 to zero or to hv. 



In vacuo 



_ f* 1 ds 



JQ VflJj X ' 



and the necessary and sufficient condition of tautochronism is 

 s 2 = Ax. Hence generally 



fi dz 



J V.FZ FZ 



is independent of z l9 provided z* = A . Fz. Now, when R = lev*, 

 there is 



vdv , o dx 



therefore 



= - 2^ f e^ ~ &. 

 ^J ds 



therefore er**t?* = 2^ (^ fs), 

 and 



Put ^ = e"* 8 ^, and let 2? and 5 be equal to zero together, 



therefore z = i (1 - r*) , fs = -P, 

 * Cambridge Mathematical Journal, No. X. Vol. n. p. 153, November, 1840. 



