xxix.] EXCEPTIONAL PHENOMENA. 651 



but when the media differ in refractive power the less 

 angle cannot become a right angle, as this would require 

 the sine of an angle to be greater than the radius. It might 

 seem that this is an exception of the kind described below 

 as a limiting exception, by which a law is shown to be in- 

 applicable beyond certain limits; but in the explanation 

 the exception according to the undulatory theory we 

 find that there is really no breach of the general law. 

 W hen an undulation strikes a point in a bounding surface 

 spherical waves are produced and spread from the point 

 Ihe refracted ray is the resultant of an infinite number of 

 such spherical waves, and the bending of the ray at the 

 common surface of two media depends upon the compa- 

 rative 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 

 omet. 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 

 3ase 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. 



