OF DEFLECTIVE FORCES. 139 



281. The equation of the tautochronous 

 curve, in a resisting medium, is 7igdz=kds 

 (1 — e—''^) ; g being the force of gravity, z the 

 vertical ordinate, s the length of the curve 

 from the lowest point, and k a constant quan- 

 tity : the resistance being expressed by m 



ds . ds« 



The forces acting on the moving point are, first, the 

 force of gravity reduced to the direction of the curve, 

 which is expressed by ^ -r- ; and secondly r, the resistance 



of the medium, which depends in general on the velocity 



d* 



■J— : and it follows from the definition of an accelerating 



force, th|it the fluxion of the velocity is its measure, (228, 



229), consequently, in the ascentof the body, — dv=g— + r, 



and 0=d -^+ or — -fr, or, making d^ constant, 0=-7--4- 

 d^ ds dp 



dz 

 g —-^Ty which is more circuitously expressed in the ori- 

 ds 



ginal notation 0= — +g--+<P (^), (i\ r being called 



d* 

 a " function" of —-. The notation is, however, imme- 

 d^ 



diately exchanged for the more convenient supposition of 

 a resistance proportional to the sum of two powers of the 

 Telocity, (p (— --)being w* -p+ n _ . We must now assume 

 a variable quantity m, dependent on x, and making p=: 



