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LVI. p. 1. Phil. Soc. vol. xiv. p. vol. xvi. p. 171. dv\ du dv dw See also Hill, Quart. Journ. vol. xvn Tram. Camb. p. 363; Proc. Loud. Math. Soc. 1; Phil. Trans. 1884, ; CLEBSCH'S TRANSFORMATION. H=-Q + j Putting 29 + \j + tf ............... (16), and substituting the values of du/dt and dQ/dx from (15) and (16) in (]), we obtain with two similar equations. Multiplying by f, 77, f and adding, we obtain dH = dH --h dH --t-c-7 * f-7 ^ r) ass ............... (18). P, = .................. (20). dt = A, which = whence the 9^/9^ and X Similarly therefore the vortex lines are always composed same elements of by Helmholtz established Hence =A = 0.

2 ) x ~ ( ~ 6) a) ) V 3 8 2 and two similar equations: where u, v, w are the component velocities of the fluid relative to the moving axes whose component to X - - i angular velocities are A 10. a) l} o> 2 o> , 3 . solid sphere of radius a surrounded by a mass of is 3 liquid whose volume is 47rc /3, and its centre is a centre of attractive force varying directly as the square of the distance. If the solid sphere be suddenly annihilated, show that the velocity of the inner surface when its radius is #, is i *V {(of 4 cT where p co} = given by /9TT 9 + \ p?

Integrating along a stream line, and restoring the values of Q and T, we obtain Now if dt p F(f) dt . 36. The equations of motion may be deduced, as Mr Larmor has shown, by means of the Principle of Least Action combined with the Lagrangian method. Let x whose y z be the coordinates at time y, initial coordinates are a, b, c t of an element of fluid the Principle of Least Action ; requires that //// should be a { V (** +? + *) - Vp] dt dxdydz 2 maximum or minimum d subject to the condition that ' p (a, b, c) where the time of the motion is constant.

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A Treatise on Hydrodynamics, Vol. 1 by A. B. Basset


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