With a given shape and size of body, equiôÙlibôÙrium in the tube may be maintained under greater radial pressure towards one end than towards the other. For example, a cylinder having conical ends, of semi-angles ö¡ and ö¡ÿ£¢ãý respectively, remains in equiôÙlibôÙrium, apart from friction, if pÿ£¢cosÿ£¢2ã₤ö¡ =ã₤pÿ£¢ãýÿ£¢cosÿ£¢2ã₤ö¡ÿ£¢ãý, so that at the more tapered end where ö¡ is small p is small. Therefore the whole structure might assume such a configuration, or grow under such conditions, finally becoming rigid by solidification of the envelope. {658} According to the preceding paragraph, we must assume some initial distribution of pressure, some squeeze applied to the posterior part of the egg, in order to give it its tapering form. But, that form once acquired, the egg may remain in equiôÙlibôÙrium both as regards form and position within the tube, even after that excess of pressure on the posterior part is relieved. Moreover, the above equation shews that a normal pressure no greater and (within certain limits) actually less acting upon the posterior part than on the anterior part of the egg after the shell is formed will be sufficient to communicate to it a forward motion. This is an important consideration, for it shews that the ordinary form of an egg, and even the conical form of an extreme case such as the guillemotãs, is directly favourable to the movement of the egg within the oviduct, blunt end foremost.
The mathôÙeôÙmatôÙiôÙcal statement of the whole case is as follows: In our egg, consisting of an extensible membrane filled with an incompressible fluid and under external pressure, the equation of the envelope is pÿ£¢nã₤+ã₤T(1ã₤ãã₤rã₤+ã₤1ã₤ãã₤rÿ£¢ãý) =ã₤P, where pÿ£¢n is the normal component of external pressure at a point where r and rÿ£¢ãý are the radii of curvature, T is the tension of the envelope, and P the internal fluid pressure. This is simply the equation of an elastic surface where T represents the coefficient of elasticity; in other words, a flexible elastic shell has the same mathôÙeôÙmatôÙiôÙcal properties as our fluid, membrane-covered egg. And this is the identical equation which we have already had so frequent occasion to employ in our discussion of the forms of cells; save only that in these latter we had chiefly to study the tension T (i.e. the surface-tension of the semi-fluid cell) and had little or nothing to do with the factor of external pressure (pÿ£¢n), which in the case of the egg becomes of chief importance.
The above equation is the equation of equiôÙlibôÙrium, so that it must be assumed either that the whole body is at rest or that its motion while under pressure is not such as to affect the result. Tangential forces, which have been neglected, could modify the form by alteration of T. In our case we must, and may very reasonably, assume that any movement of the egg down the oviduct during the period when its form is being impressed upon it is very slow, being possibly balanced by the advance of the {659} peristaltic wave which causes the movement, as well as by friction.
The quantity T is the tension of the enclosing capsuleãthe surrounding membrane. If T be constant or symmetrical about the axis of the body, the body is symmetrical. But the abnormal eggs that a hen sometimes lays, cylindrical, annulated, or quite irregular, are due to local weakening of the membrane, in other words, to asymmetry of T. Not only asymmetry of T, but also asymmetry of pÿ£¢n, will render the body subject to deformation, and this factor, the unknown but regularly varying, largely radial, pressure applied by successive annuli of the oviduct, is the essential cause of the form, and variations of form, of the egg. In fact, in so far as the postulates correspond near enough to actualities, the above equation is the equation of all eggs in the universe. At least this is so if we generalise it in the form pÿ£¢nã₤+ã₤Tã₤ãã₤rã₤+ã₤Tÿ£¢ãýã₤ãã₤rÿ£¢ãý =ã₤P in recognition of a possible difference between the principal tensions.