APPLICABLE TO THE MOTION OF FLUIDS. 375 



dx _ u dy _ v d% _ w 

 ds~~V' ds~~T' ds~T'' 



so that, ^ = 4<> 2 + »* + «0 = F", 

 as V 



rfs 2 efo rf* 2 c?y* da? 



Hence by substituting in the foregoing value of — | — T , we obtain, 



1 + 1- llt^ + m + ^l) v *_v* d JL\. 



r r' " r 3 \W + <fy 2 ^ dz*) ds]' 



dV p-fl 2.V- ^ 4- ^ ^ ^ ^ - ^ M ^ v — 

 ' ds \r // " da? dy 2 dz' 2 ~ dx dy dz' 



The transformed equation sought for is therefore, 



£+'£**)-> «• 



4. A similar transformation of equation (1) may readily be effected 

 when p is variable. For this equation may be put under the form, 



dp dp dp dp du dv dw 



pdt pdx pdy pdz dx dy dz 



Hence putting ^r for u, -j- for v, and -=-? for w, we have 



dp 1 (dP) du dv dw _ 

 pdt of' dt dx dy dz 



But it must be observed that the variation in {dP) is from one point 

 to another in the line of motion, on account of the above substitu- 

 tions for u, v, and w. 



Hence , ' = V , , = Vd* .—jj-^. Also, as has already been proved, 

 dy 



du dv dw _ dV „ (1 1\ 

 dx dy dz ' ds \r r'J * 



T T 2 



