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By A. B. Basset
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Extra resources for A Treatise on Hydrodynamics, Vol. 1
43). -f or y t right-hand side must be z v Since the left-hand side of (43) c, 37 a perfect differential, the be a function of b and is so also, whence B= dF > if F we must have C 7? = dF Q- T = F(b, and therefore is ( ...................... (45) c) an integral of the equations of motion. and When C are 40. the motion is irrotational, We must now and therefore lt s a an absolute constant. , find the values of w and stitute the values of u, v and differentiate M M M F is each zero, and therefore we partially, B and G.
2T=u* + v2 + w*, Putting and multiplying equations (25) by dx, dy, dz respectively and adding, we obtain M= where v(vx l - uy) + w(u M with similar expressions for it follows that 2 3 2 M M wx )= z and M 3 2v From . + '~2wrj, the values of M lt , Eliminating u, v, w from (39) and M I} bx} cx (41), = we obtain 0. Hence we may put M^Bbl+Cc'A where B determine and ; C (40) are quantities whose values may now be 1 written Crelle, vol. LIV. p. 293. (42), we shall hereafter CLEBSCH'S TEANSFOKMATION. dQ-dT = B (b x dx b dy f b dz) + C (c xdx + c dy + c dz), dQ-dT=Bdb + Cdc .....................
Fluid, and let p, p be the upon ABC By D'Alembert's Principle, the reversed effective forces and the impressed forces which act upon the volume of fluid, together with the pressures upon its faces, constitute a system in statical The first two vary equilibrium. as the volume, and the last vary as the areas of the faces of the tetrahedron ; and therefore if the tetrahedron be made to diminish former will vanish in comparison with the latter. Hence the tetrahedron will ultimately be in equilibrium under the indefinitely, the action of the pressures upon its faces.