185, 186] EQUATIONS OF MOTION. 497 



7 d*x -. du 



gdr w = 9 dr jt , 



o\ ^ d^u -, dv 



2) (rdT-^^edr-^, 



^ d*z 7 dw 



Q dr^ = Q dr- d ~, 

 and these are to be subtracted from the applied forces 



and introduced in 1). Consequently we have the equations of motion 



du\ dp 



(x- 



(z- 



dw \ dp 



~dt I ~" Js 



Now by the ordinary derivative ^- is meant the rate of change 



of velocity of a particular particle as it moves about. If we have 

 any function F pertaining to a particular particle we may write its 

 derivative 



.^ dF = dF ^dFdx ,dF<h , gg jf " 



dt ~~ dt ' -dx ~dt + dy ~di + ~dz Tt' 



f)F 

 where -^ would be the rate of change of F if the particle were at 



rest. The derivatives > -j-i ~ are the velocity components of the 

 particle, u, v, w. Accordingly we have 



KX dF 2F , dF , dF , dF 



5) ^T~ == ^T + ^O h v Q h^^~- 



dt dt dx ' dy dz 



We shall call this mode of differentiation particle differentiation. 1 ) 



Introducing this terminology, dividing by Q and transposing, our 

 equations of motion 3) become 



DF 



1) In most English books the symbol > used by Stokes, is used for 



jj t 



particle differentiation because of the very objectionable practice of making no 

 distinction in the symbol for ordinary and partial differentiation. 



, Dynamics. 32 



