Condition of the Rectilinear Motion of Fluids. 105 



Hence, cos <a»ff — cos < APQ = — — - — - — ' =., 



tH ar(l + r)^l+a n -\-y r 



We have, therefore, by obvious substitutions, 



V(aa!r-l) V,aa! 



dw _ r (r + I) ij 



dx •! + a 9 + #"."• 1 ,+y* + V*' 



So if V 2 , r 2 be the increments of V and r, the coordinate z 

 only being supposed to vary, by exactly the same reasoning 

 we shall obtain, 



V(aa'r — l) V^a'a 



du _ r(r + 1) r 2 



dz~ V 1 + «*"+ b n . V 1 + « 2 + 6 2 ' 



If, therefore, — = -=— , we must have — - = — - • 

 dz d.z r 2 rj 



Hence, V 3 and r 3 being corresponding increments of V and 

 r when^/ only varies, we may conclude that 



. f du _dv du_dw ^dv_d'W t 



dy ~ dx 9 dz~ dx* dz dy* 



that is, if u d x + v dy + na dz be an exact differential. As- 

 suming now that r x = r 2 == r 3 , we shall also have \, = V 2 = V 3 . 

 Hence the increments of velocity in the directions of the axes 

 of coordinates are the same, when the projections of the 

 increments of the coordinates on the line of motion are the 

 same. As the directions of the axes of coordinates may be 

 arbitrarily assumed, the general inference from this result is, 

 that when udx + vdy+wdz is an exact differential, the 

 increment of velocity from one point to another at a given 

 time depends only on the change of position in the direction 

 of the motion; which it was required to prove. 



Supposing now that u dx -f vdy + ivdz = d <£, that p is 

 the pressure and g the density at the point x y z, and that 

 X, Y, Z are the impressed accelerative forces at that point in 

 the directions of the axes of coordinates, we have the known 

 general equation, 



f l lP=f { Xdx + Ydy + Zdz)- d £-^+f(t), 



which, being differentiated with respect to space, gives 



^ = Xdx + Ydy + Zdz-d. d -£-VdV. 

 g * dt 



