1£8 CELESTIAL MECHANICS. I. ii. 10. 



dx constant, and df of course variable, da-r— -:=:C -r— d^ 



dt' as ^ 



, ^ dsddt , ^ ddz dzdd^ . dz.^ .^ 

 and S = -— ; also = — — ^ — + f j^df -^d«; 



consequently ^yd^^ ~ ddz Tr~ + ^TT^' • ""* "JT^ 



d^ d* df 



ds 



Cd/2 



—7— , so that gdt^=id^z, and, taking the fluxion, 2^d^d2< 



^d<3 ^ ^ ^ ddz 



tzd^z ; now since d^^ = — : — , and dt^zz. , we have d'z 



as g 



2^d^z2 





d* ds ds ^ g ^ yds 



Q dsd^z 

 and - — -r-;m;, which determines the law of the resistance 

 g 2d>z- 



Qf required for the description of a particular curve. 



Now supposing the resistance proportional to the square 



of the velocity, which is nearly true in a medium of 



12 



uniform density, Q being expressed by A- t-^, we have 



Q L~ 



Q hds"- /hU-2 , , , d^z . Ms"^ Q d.sd"z 

 — = — ->——:r-y and lidszz — r-r-, smce-rT-==: — = 77-. — ; 

 g gat- d^z 2ddz ddz g 2d^z" 



hence, taking the fluent, we have hs-=z^ l)ld"z + c, or 2/t5:= 



hld^z + t, in which, since do: is constant, we must take 



ddz 

 l2zH-c=hl 



<5dz 

 da^-''^ • 



Corollary. If we make Ji-=.Oy and suppose the 

 resistance to vanish, we have d-z=i«da^; the fluent of 

 which is dzz=.^axdx -\-hdx, whence zzzirtx-H-ix + c, which 

 is the equation of a parabola (204) h and c being deter- 

 mined by the conditions of projection ; and since d^zzz 



hld"zH-c=:hl— :— , and since hi (ae '^'^^) =27^5 -fhla, we have 



