DISCONTINUOUS FUNCTIONS. 239 



between given limits, and the function of x to 

 vary from one function to another, during the 

 variation of x, so that no function of c€ shall cor- 

 respond to two values of ,r, however near to each 

 other ; we shall then have a species of variation 

 which corresponds to the calculus of variations, 

 an example of which was first given by Sir I. 

 Newton, in the Principia, book 2, prob. 34. 



(6). If we suppose the function of x to vary, 

 while X takes the values 1, 2, 3, &c. we shall have 

 a species of variation which might be, with pro- 

 priety, denominated the variation of the calculus 

 of finite diSerences. Many instances of the 

 application of such a calculus might be given, one 

 of which is the following : 



Let A and B be two points a 

 not in the same vertical line; 

 and suppose a material body 

 to move from A to B, by the 

 force of gravity, along n in- 

 clined planes, the length of ^~~--..__ b 

 each being a ; Required the position of the planes 

 when the material body moves from A to B in 

 the least time possible ? 



