458 Mr. POWER, ON A THEOREM IN FLUID MOTION. 



If, then, the preceding expression be developed in powers of t, the 

 coefficients of the different powers of t will severally be exact differ- 

 entials with respect to x, y, %*. 



Let u, v', w, denote the value of u, v, w when t = 0. Since in all 

 cases of nature the velocities u, v', w' must be finite, it follows that the 

 general values u, v, w will be developable in ascending positive powers 

 of /, integral or otherwise, and the smaller t is taken, the more accurately 

 will these general values be represented by the earlier terms of the 

 series, and it is quite sufficient for our purpose to regard t as indefi- 

 nitely small. 



We may therefore assume 



u = u + u"P + u'"F + &c, 

 v = v' + v"t + «'"<* + &c, 

 w = w' -j- w"t K + w'"t» + &c, • 



* Let L denote any exact differential with respect to x, y, z, L at the same time con- 

 taining t. 



Suppose that, expanding L in powers of /, we have 



L = £,,<" + tj + L 3 f + &c, 



where a, /3, y, &c, proceed in ascending order from the negative to the positive infinity. 



Since the right-hand side is by hypothesis a complete differential with respect to x, y, z, 

 whatever be the value of t, it will continue so when divided by /", 



therefore £, + L s t^~ a + L 3 ^" a + &c. 



is so for all values of t, and consequently when t = 0. Therefore Z., is an exact differential 

 (for since ft — a, y - a, are all positive, the remaining terms vanish when t = 0). 



Hence also L s t^~ a + L a f~ a + &c, 



and L a + L B t y ~ l> + &c, 



are exact differentials for all values of I, and therefore when 1 = 0. 



Consequently L a is so. 



In the same way it may be shown that L a , L t are all exact differentials with respect 

 to x, y, z. 



If the developement be supposed to contain a term of the form Mi™ (log, t)", it may, 

 if necessary, be demonstrated in a similar manner that M is an exact differential with 

 respect to x, y, z. 



