OF DEFLECTIVE FOR.CES. 143 



limits of ^1=0 and tzzT. We must therefore make ku 



alone equal to ^---^^^^—-^ and k (e"^~\)—^—-, whence 

 p-dii pas pds 



kds (e"* — l):=zgdz. — =^dz.we"% and ngdz^^kds (1— e"'"). 



282. Corollary. When the resistance 

 either disappears, or is proportional to the ve- 

 locity only, /i=0, and the equation becomes 

 gdz=^ksds, which belongs to the cycloid. 



For since e-^^=:l — ns-] — ;r-+ . . .(247, Cor. 3), when /* 



/ml 



vanishes, 1 — e-'^^ziins, and ngdzz=.nksds, [Tliis equation 

 is shown to belong to the cycloid in article 287.] 



Scholium 1. It is remarkable that the coefficient n 

 of the part of the resistance proportional to the square of 

 the velocity does not enter into the expression of the time 

 T; and it is obvious, from the steps of the analysis, that 

 the expression would be the same, if we added to the pre- 

 ceding law of the resistance terms proportional to the 



ds^ ds'^ 

 higher powers of the velocity — , -rT4 • • • [That k is inde- 

 pendent of w, appears from making s very small, when ngdz 



sds 

 zznksds,axid k=. —p, whether n be greater or smaller.] 



Scholium 2. " In general, if the retarding force in 



the curve be U, we shall have 0= ^ ^R, the space s be- 



ing a function of the time t and of the whole arc described, 



which is of course a function of t and s ; and by taking 



the fluxion of this last function, we may obtain an equa- 



d* 

 tionofthe form— =:F, the velocity being thus repre- 



