DISCONTINUOUS FUNCTIONS. 251 



Let AC Fig. (5) represent y, ^ 

 the solutions of the indetermi- 

 nate equation j^ ~ f (■^■) foi' every 

 value of a; between the limits 

 .1- zz and x zz AB. This cor- 

 responds with Descartes' view of 

 the subject, and supposes the function of .v to 

 remain the same for every value of .v. Now, if 

 we suppose the function of .r, namely f(aj), to vary 

 for every value of ^ between the limits .v = o 

 and X zz AB. The examination of the solution 

 of each function for one or more values of ,i' 

 between the limits assigned above, is what I 

 have ventured to suggest in the former part of 

 this article. 



(13). In most physical researches, the great 

 object has been hitherto to discover a constant 

 law that will connect the effects which are observed 

 with the cause assigned to produce them. Those 

 only who have tried properly know the difficulty 

 of these speculations; and the few who have suc- 

 ceeded in their attempts to give a constant law 

 connecting natural phenomena with their causes 

 appear, to me at least, to have been successful 

 only between assignable limits. 



