Mr. POWER, ON A THEOREM IN FLUID MOTION. 459 



the indices X, /u, &c, being all positive and arranged in ascending 

 order. 



It may be observed that the indices are assumed the same in all 

 the three series, which is allowable. For, the series, as exhibited above, 

 may clearly be made to embrace any assignable case by causing to vanish 

 one or more of the coefficients u", v", w", u", &c. Thus, X being the 

 lowest index which occurs in any one of the three series, if it occurred 

 only in the first, we should have v" = 0, w" = ; if it occurred in the 

 first and third only, we should have v" = 0, and so on. The same may 

 be said of n, the next lowest index which occurs in any one of the 

 three series, and so on for the other indices. But the evanescence of any 

 of these coefficients will not affect the following reasoning. Hence the 

 legitimacy of the assumption is manifest. 



If we substitute the developements of u, v, w, in the expressions of 

 «> ft 7» we find 



a = a + « *. + a'P + &C, 



/3 = p + /S"«* + &"r + &c, 



7=7' + 7"^ + y'"^ + &c, 

 _v_. . du' dv' „ du" dv" 



&c. 



+ a(vdx — udy) + j3(wdx — udz) + y(wdy — vdx), 



