and the bending of the ray at the common surface of two media depends upon the comparative velocities of propagation of the undulations in those media. But if a ray falls very obliquely upon the surface of a rarer medium, the waves proceeding from successive points of the surface spread so rapidly as never to intersect, and no resultant wave will then be produced. We thus perceive that from similar mathematical conditions arise distinct apparent effects.
There occur from time to time failures in our best grounded predictions. A comet, of which the orbit has been well determined, may fail, like Lexell’s Comet, to appear at the appointed time and place in the heavens. In the present day we should not allow such an exception to our successful predictions to weigh against our belief in the theory of gravitation, but should assume that some unknown body had through the action of gravitation deflected the comet. As Clairaut remarked, in publishing his calculations concerning the expected reappearance of Halley’s Comet, a body which passes into regions so remote, and which is hidden from our view during such long periods, might be exposed to the influence of forces totally unknown to us, such as the attraction of other comets, or of planets too far removed from the sun to be ever perceived by us. In the case of Lexell’s Comet it was afterwards shown, curiously enough, that its appearance was not one of a regular series of periodical returns within the sphere of our vision, but a single exceptional visit never to be repeated, and probably due to the perturbing powers of Jupiter. This solitary visit became a strong confirmation of the law of gravity with which it seemed to be in conflict.
Singular Exceptions.
Among the most interesting of apparent exceptions are those which I call singular exceptions, because they are more or less analogous to the singular cases or solutions which occur in mathematical science. A general mathematical law embraces an infinite multitude of cases which perfectly agree with each other in a certain respect. It may nevertheless happen that a single case, while really obeying the general law, stands out as apparently different from all the rest. The rotation of the earth upon its axis gives to all the stars an apparent motion of rotation from