APPLICABLE TO THE MOTION OF FLUIDS. 381 



borne in mind that on account of these substitutions, the variation in 

 (d<p) is from one point to another in the line of motion. 



10. Resuming the fundamental equation (3), and substituting in it 

 the values of u, v, w, we have 



/d.Np\ (d.N*£\ (d.NipK 



Id d ^\ 



at ( dx \ _ d*<p d*(p dx d*<p dy d 2 <f> dz 

 \ dt / dx dt dx 2 ' dt dx dy ' dt dx dz ' dt 



d*(p „ (d$ (Ftp d<p d 2 <p d(p d 2 cf> \ 

 dx dt \dx ' dx' dy ' dx dy c?a ' dx dz ! ' 



(d.f\ (d d £\ 



and similarly for 1 - _ . / and I — -* — / . 



Hence by performing the differentiations of the foregoing equation, 

 the result expressed in the notation already used, will be 



Therefore by integration, 



The equation (13), makes the last term disappear. 



11. If N be a function of t only, 



(dN) - r dN , , s dN 



Vol. VII. Paet III. Uu 



