EXCEPTIONAL PHENOMENA. 315 



would require the sine of an angle to be greater than the 

 radius. It might seem, perhaps, that this was an exception 

 of the kind elsewhere described as a limiting exception, in 

 which a law is shown to be inapplicable beyond certain de- 

 fined limits ; but in the explanation of the exception 

 according to the undulatory theory, we find that there 

 is really no breach or exception to the general law. 

 Whenever an undulation strikes any point in a bounding 

 surface, spherical waves are produced and spread from 

 the point. The refracted ray is the resultant of an infi- 

 nite number of such spherical waves, and the bending of 

 the ray at the common surface of two media depends upon 

 the comparative velocities of propagation of the undula- 

 tions in those media. But if a ray falls very obliquely 

 upon the surface of a rarer medium, the waves arising 

 from successive points of the surface may spread so rapidly 

 as never to intersect, and no resultant wave will then be 

 produced. We thus perceive that from general mathe- 

 matical conditions may arise very distinct apparent effects. 

 There may occur from time to time distinct failures in 

 our most well-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 hold 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 gravi- 

 tation itself deflected the comet. As Clairaut remarked, 

 in publishing his calculations concerning the expected re- 

 appearance 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 influ- 

 ence of forces totally unknown to us, such as the action 

 of other comets, or even of some planet too far removed 

 from the sun to be ever perceived by us. In the case of 



