APPLICABLE TO THE MOTION OF FLUIDS. 377 



The arbitrary function of the time F' (t) may be got rid of by 

 substituting <p' + fF{t)dt for <p. 



Again, by differentiating equation (8) with respect to s, an equation 

 of the same order as the preceding may be obtained, having V for 

 principal variable. The result (since dr = dr 1 = ds) will be, 



*Q dV dQ d»Q dT d*V dF> 



dsdt ds' ds + ds 2 { } d# df ■ ds* 



~rr d*V n dVdV a dV i\ 1\ _ _/l in 

 " ^ds-dt - 2 - ds-dt + a ' ds (r + ?) ~ ra {? + V*) = °-- (10 > 



It should be observed that equations (9) and (10) are subject to the 

 same limitation as equation (5) in regard to the direction of the variation 

 of the co-ordinates. 



6. All the preceding results have been obtained on the supposition 

 that udx + vdy + wdz is an exact differential. It will now be proper 

 to enquire to what circumstances of the motion this analytical condi- 

 tion refers. The following considerations will enable us to do this. 



It has already been proved in Art. 3 that, 



udx + vdy + wdz = Vdr\ 



in which dr may be considered the increment of a straight line drawn 

 in the direction of the motion, and dx, dy, d% are the corresponding 

 increments of the co-ordinates, the variations taking place at a given 

 instant from one point to another indefinitely near. Now Vdr is not 

 an exact differential unless V may be considered a function of the line 

 r ; that is, unless the variation of V from one point of space to another 

 at a given instant depends only on the change of position in the direc- 

 tion normal to the surface of displacement, the variation from one point 

 to another of the surface of displacement being zero. Therefore also 

 udx + vdy + wd% is not an exact differential unless dV= when the 

 co-ordinates vary at a given instant from one point to another of a given 

 surface of displacement.* 



* See a direct proof of this Proposition in the ' Note' added to this Paper. 



