Recalling the formulae above which connect s P4 and a m, we see that **dP**4 and **Dp** q are in co-relation with these quantities respectively, and may be said to be operations which correspond to the partitions (pq), (10 P 01 4) respectively.

But by Green's transformation f flpdS = f f PPdxdydz, (2) thus leading to the differential relation at every point = dy **dp** The three equations of equilibrium obtained by taking moments round the axes are then found to be satisfied identically.

Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =f**dp**lp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then **dP** dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.

From the gas-equation in general, in the atmosphere n d **dp** _ I **dp** 1 de _ d0 de i de (8) z p dz-edz-p-edz-k-edz' which is positive, and the density p diminishes with the ascent, provided the temperature-gradient de/dz does not exceed elk.

With uniform temperature, taking h constant in the gas-equation, **dp** / dz= =p / k, p=poet/ k, (9) so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p 1 and 1, 2 at heights z 1 and z2 (z1-z2)11?