t 



OF DEFLECTIVE FORCES. 127 



Resuming the equation (f, 264), 0=Sx(d -77— P<it)-i- 



gy (d-~— §dO + Sii (d-j-— i2dO, and supposing z to 



begin at the highest point of the curve, we may resolve the 

 force of resistance C into three directions, and it will 



afford us — C -r^, — C^^, and — C -r- ; consequently P 



X, d^' ^ ^ d?/ , „ ^ dz __ 



=: — ^T-, 0=— ^T^» and jR = — ^3- + 2-. Hence we 

 as ^ ds d* ^^ 



haveO=8.(d^5f+ fgdO + S,(dJ+egdO + gz(dJ +C 



dz 



j-dt^gdt) : and if the motion is subjected to no further 



.... , 1 . r. 1^^ ^ dx 



limitation, we have the three equations Ozzd-T— + t j- 



df; 0:=d-;^ + ^7^d^; and = d :tt + ^ J-d^— ^df. From 

 d^ ds dt ds ^ 



the two first, we obtain, by multiplication and subtraction, 



dif dx dx dij 



jj. d -r-zzj-. d-j^; and, dt being constant, dividing both 



.. , dxdy ^ ddx ddy . ,,, ,,^ 

 sides by . ' , we have-; — ^—~t and hldx=:hidv + c~hl 

 at^ ax ay "^ 



fdy, and dxzi/dy,/' being a constant quantity. But since 



dxzzfdyy the horizontal motion must be rectihnear, and 



the body must move in a vertical plane, which is indeed 



sufficiently obvious from the absence of any lateral force. 



We may, therefore, consider x as situated in this plane, y 



dx 

 being always =0; and from the two equations Ond— + 



^ dx dz dz 



^f fe J~*^^* ^^^ ^ ~ ^T7 + ^ JT^^^""^'*^^' w® obtain, making 



