IN HYDRODYNAMICS. 39 



given element of the fluid, but which are constant in passing from point to point of a surface 

 of displacement through either a finite or an indefinitely small space. Let, therefore, the 

 equation \^ = be equivalent to f{,v,y, z, a, b,c,kc.) = 0. Then 



d\l/ df da df db df dc 



— I-=-A. — + ^. — +-^. + &C. 



dt da dt db dt dc dt 



dyb df df da dx df db d.v df dc dw 



M-^ = M— +— . — . — +— . — . — +— . — . — + &C. 

 d.v dx da dx dt db dx dt dc dx dt 



J I -It 



and so for w — and u—^. Hence, substituting in equation (2), we have 

 dy dz 



df lda\ df ldb\ df ldc\ df df df 



da \dtj db' \dtl dc ' \d(J ' da; dy dx 



Now differentiating the equation f{x, y, z, a, b, c, &c.) = 0, we obtain, since a, b, c, &c. are 

 constant for a given surface of displacement, 



df , df , df , u , V , w , 



— dx+ — dy + -fdz=-0=~dx + ---dy + —-dz. 

 dx dy dz N N N 



Hence, u=N-f-, v = N~, w = N~: and consequently, 

 dx dy dz 



df /da\ df (db\ df /dc\ „ ,, / dp df- df'\ 



£■ [diJ-'i-U^ic- U) +^'^- + ^(£^ + 57-^^0 ^'^ ^'"^- 



We shall presently illustrate the use of this equation in finding A''. 



It is plain that the equation f(x, y, z, a, b, c, &c.) = 0, may be that of a surface having a 

 contact of the second order with the surface of displacement at any point xyz, the parameters 

 in the equation of such a surface being a, b, c, &c. For instance, let the equation of the surface of 

 contact be 



(^-»Y ^ (y - ^y ^ (^ - 7)° , „ 



w? n' p' 



then we have for determining A'^ the general equation, 



g-a lda\ y-(^ /dfi\ z-y /dy\ (x - af ldm\ (y - fif tdn\ {z - 7)^ ld±\ _ 

 ni' '\dtl^ rv" '\dtl p' '[dt] m? '\dtl'^ v? '\dt)'^ f ' \dt] ~ 



V m' ^ n\ ^ p' J 

 I proceed now to adduce some examples of finding i\r, and of applications of the equations 

 (8), (9), and (:0), to determine whether proposed instances of motion are possible. 



Ex. 1. Let the case of motion be that considered in Art. 3. This instance gives u = mx, 



, ,. ^ , . dti dv 



v=-my, and satisfies the equation — + _ = 0. Also udx + vdy = m{xdx - ydy), and 



u dy y , 



- = 7- = • Hence the general equation of the surfaces of displacement may be assumed to 



be x' - y" - a' = 0, and the general equation of the lines of motion, xy = c". 



