620 THE PRINCIPLES OF SCIENCE. [CHAP. 



Secondly, it must be remembered that mathematical laws 

 of some complexity will probably present singular cases or 

 negative results, which may bear the appearance of discon- 

 tinuity, as when the law of refraction suddenly yields us 

 with perfect abruptness the phenomenon of total internal 

 reflection. In the undulatory theory, however, there is 

 no real change of law between refraction and reflection. 

 Faraday in the earlier part of his career found so many 

 substances possessing magnetic power, that he ventured on 

 a great generalisation, and asserted that all bodies shared 

 in the magnetic property of iron. His mistake, as he 

 afterwards discovered, consisted in overlooking the fact 

 that though magnetic in a certain sense, some substances 

 have negative magnetism, and are repelled instead of being 

 attracted by the magnet. 



Thirdly, where we might expect to find a uniform 

 mathematical law prevailing, the law may undergo abrupt 

 change at singular points, and actual discontinuity may 

 arise. We may sometimes be in danger of treating under 

 one law phenomena which really belong to different laws. 

 For instance, a spherical shell of uniform matter attracts 

 an external particle of matter with a force varying inversely 

 as the square of the distance from the centre of the sphere. 

 But this law only holds true so long as the particle is 

 external to the shell. Within the shell the law is wholly 

 different, and the aggregate gravity of the sphere becomes 

 zero, the force in every direction being neutralised by 

 an exactly equal opposite force. If an infinitely small 

 particle be in the superficies of a sphere, the law is again 

 different, and the attractive power of the shell is half what 

 it would be with regard to particles infinitely close to the 

 surface of the shell. Thus in approaching the centre of a 

 shell from a distance, the force of gravity shows double 

 discontinuity in passing through the shell. 1 



It may admit of question, too, whether discontinuity is 

 really unknown in nature. We perpetually do meet with 

 events which are real breaks upon the previous law, though 

 the discontinuity may be a sign that some independent 

 cause has come into operation. If the ordinary course of 



1 Thomson and Tait, Treatise on Natural Philosophy, vol. i. pp 

 346351, 



