The theory of the equilibrium and motion of fluids / By Thomas Webster.

  • Webster, Thomas, 1810-1875.
Date:
[1836]
Catalogue details

Licence: Public Domain Mark

Credit: The theory of the equilibrium and motion of fluids / By Thomas Webster. Source: Wellcome Collection.

    22/238 (page 10)
    12. The conditions of equilibrium which we have obtained, require that p should be a function of x, y, z, such as will satisfy the three equations (l) at once, or satisfy (2). Hence, that the value of p may be possible, the product of p and Xdx + Ydy + Zdz must be a com¬ plete differential of some function of the three independent variables a?, ?/, z. Conversely, when this product is a complete differential of a functiou of these variables, the value of p can be found by integration, and the three equations (l) will be satisfied. This then being the case, we have p=fp (Xdx + Ydy + Zdz).(3). This integral may be taken with regard to any series of consecutive values of a?, y, z. Hence, the integral taken with regard to any such series of values, or in other words, the pressure of every line of fluid particles or canal leading to the same point, is the same; this was assumed by Newton as the basis of the theory of the equilibrium of fluids, and would lead at once to the preceding conditions. The principle that we may integrate in regard to any line whatever of fluid particles drawn from the point to the free parts of the fluid admits of some important appli¬ cations, as will be seen hereafter. 13. That the equilibrium may subsist, we must have p (Xdx -f Ydy + Zdz) a complete differential, which it is when d . p X d . pY dy d . p X doc ’ dz dX d d • jO i doc or wp dY + x- = p~,—f * dy dy dx d . pY dz dp d . pZ dy dX vdp p —-b X - dz dY dz di dx dZ + Z dx ’ dp dx 5 dp dy ' dy ’ iET* di? f d'f- {j e k Ov)r . ** 0 ‘a L» 4 *