Theorem. Let the accelerating forces X, Y, Z, acting on the fluid, be such that Xdx + Ydy + Zdz is the exact differential d V of a function of the coordinates. The function V may also contain the time t explicitly, but the differential is taken on the supposition that t is constant. Suppose the fluid to be either homogeneous and incompressible, or homogeneous and elastic, and of the same temperature throughout, except in so far as the temperature is altered by sudden condensation or rarefaction, so that the pressure is a function of the density. Then if, either for the whole fluid mass, or for a certain portion of it, the motion is at one instant such that udx + vdy + wdz is an exact differential, that expression will always remain an exact differential, in the first case throughout the whole mass, in the second case throughout the portion considered, a portion which will in general continually change its position in space as the motion goes on. In particular, the proposition is true when the motion begins from rest.

Two demonstrations of this important theorem will here be given. The first is taken from a memoir by M. Cauchy, “ Mémoire sur la Théorie des Ondes, &c. ” (Mém. des Savans Strangers, Tom. i. (1827), p. 40). M.