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32)> whence by (28) .................. , ..... (33). When the motion takes place in two dimensions, the same way, arrive at the equations exactly ~= d? v d u -^ + ax dy , we shall, in 0, 32 EQUATIONS OF MOTION, 36 a integral of which first is whence by (34) 39. +/W = (37). subject of the steady motion of a liquid has been The manner by Clebsch 1 treated in the following . , y, z and t then if the denote differentiation with respect to x, y and z, we may ; suffixes evidently put u= bycz - b cy z for these values of u, v From (38) v , = bz c x w and - bxc w = bxcy - bycx z> (38), satisfy the equation of continuity.

In terms of w () ty, (30) } ^ dw)} evidently *+ + *+ w 0, id* = d^-v-d* fw - . ............... (32)> whence by (28) .................. , ..... (33). When the motion takes place in two dimensions, the same way, arrive at the equations exactly ~= d? v d u -^ + ax dy , we shall, in 0, 32 EQUATIONS OF MOTION, 36 a integral of which first is whence by (34) 39. +/W = (37). subject of the steady motion of a liquid has been The manner by Clebsch 1 treated in the following . , y, z and t then if the denote differentiation with respect to x, y and z, we may ; suffixes evidently put u= bycz - b cy z for these values of u, v From (38) v , = bz c x w and - bxc w = bxcy - bycx z> (38), satisfy the equation of continuity.

The equation of a velocity of the surface normal to itself moving surface the is + (dF/dz}\ Hence deduce equation and u, v y and z are given functions of a, 6, c and t, where a, constants for any particular element of fluid, and if are the values of x, y, z when a, 6, c are eliminated, If x, 9. 6 (19). c are and w prove analytically that d*x 10. ^ Liquid which is moving irrotationally in three dimen2 bounded by the ellipsoid (x/ctf + (yjb^ + (*/c) = 1, where EXAMPLES. a, Prove that remains constant.