fZN\ _ </N (JN) 



Differential Equations applicable to the Motion of Fluids. 89 



ence to the application I am about to make of this equation, it 

 is particularly to be remarked that the variation of the coor- 

 dinates in [d $) is from one point to another in the direction 

 of the motion. This is evident from equation (3.), by substi- 



^ . d X dy d is . ^ n i xti 



tutmg —r—^ — .-rj —TTi respectively tor u, w, and w. Now by 



the adopted notation, 



/fZN\ _ d^ 



\dt I ~ dt ' dt 



By the last substitution the limitation as to the direction of 

 the variation of coordinates is introduced. From this equation 

 it follows that 



dt dt 



Hence substituting in equation (5.), we have 



But if V be the velocity of the particle whose coordinates are 

 x,j/, z at the time t, 



_ ^ / d (p dx d (^ dy d^ dz\ 

 ~ \dx ' dt dy'dt dz ' dt )' 



ds 

 Hence V^rf^ = N (rf<f)). Or, supposing V = -^, s be- 

 ing a line drawn in the direction of the motion, V ds 

 = N {d<p). Consequently, by substitution in equation (8.), 

 we have 



This equation is known to be true by very different reasoning 

 from that employed above. 



