NOTRE-DAME DE PARIS

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1898 Excerpt: ... be immersed in the fluid. We shall consider multiply connected spaces later. 303. The truth of (47) for such cases as we have here supposed may be seen intuitively as follows, if it is admitted for a single bounding surface. Take any space, such as the space of which Fig. 66 (with A contracted to zero) may be regarded as a section, surrounding a closed space B throughout which y2 is not zero. It is only necessary to connect the space B by a narrow tubular surface with the outer boundary. This tube can be made so fine as not to interfere sensibly with the motion of the fluid. The two surfaces are thus converted into a single surface enclosing the space between them, and the surface integral, taken over the whole bounding surface, is simply the sum of the integrals taken over the two surfaces, since that over the tube is vanishingly small. When more surfaces are included it is only necessary to imagine a connecting tube for each with the outer surface (or tubes connecting all the enclosed spaces with one another, and one connecting one of these spaces with the outer surface) to enable the theorem given for a single surface to be applied. From these considerations we see that (47) asserts in such cases that the integral over the outer enclosing surface is equal and opposite to the sum of the integrals over the enclosed surfaces, that is, that the total rate of flow into the space considered over the internal surfaces is equal to that outwards across the enclosing surface, or vice versa. 1 A circuit which can be contracted to zero as here described is said to be reducible. Mean Potential over Spherical Snrface 304. Let 4ttm denote the total rate of flow into the space considered, across the inner bounding surfaces, and let the outer surface be spherical....