460 Mb. POWER, ON A THEOREM IN FLUID MOTION. 



becomes 



\t K -'(u"dx + v"dy + w"dz) + pp- l (u'"dx + v'" dy + w'"dz) + &c. 

 + {a + a"? + a"'F + &c.} {(v'dx-u'dy) + (v"dx - u"dy)t + kc.\ 

 + |/3' + P"t K + /Tr + &c.} [w'dx - u'dz + {w"dx - u"d%)t K + &c.} 

 + {y + y"? + y m r + &c.| [iv'dy - v'd% + (tv'dy - v"dz)f + kc.}, 



and, by what has been demonstrated above, the coefficients of all the 

 different powers of t are exact differentials. 



Suppose now, that udx + vdy + wdz is an exact differential when 

 t = 0, in other words 



udx + v dy + w'dz 



is an exact differential, we have then, 



du' dv 



dy dx ' 



du dw' _ 

 d% dx ' 



dv' dw' _ 

 dz dy 



that is, a =0, /3' = 0, y = 0. 



Hence it is plain that the term involving the lowest power of t in 

 the above expression is 



\t K -\u"dx + v"dy + w"d%), 

 and consequently 



u'dx + v" dy + w"dz 

 is an exact differential; whence, 



a" = 0, 0" = 0, y" = 0. 



This being the case, it follows that the term involving the next 

 lowest power of t is 



*t»-\u"dx + v'"dy + w'"dz), 



