Manchester Memoirs, Vol. xli. (1897), No. 10. 3 



In the geometrical representation, the magnitudes (1) are 

 represented by a sequence of points : — 



M l9 M 2 , M 8 , (2), 



each to the right of the preceding, but all lying to the left 

 of some fixed point A. Hence every point on the line 

 X'X, without exception, belongs to one or other of two 



Mi M 2 M3M4A X 



mutually exclusive categories. Either it has points of the 

 sequence (2) to the right of it, or it has not. Moreover, 

 every point in the former category lies to the left of every 

 point of the latter. Hence there must be some point M, 

 say, such that all points to the left of M belong to the 

 former category, and all points to the right of it to the 

 latter. Hence, if we put jx = 0M, //. fulfils the definition 

 of an " upper limit " above given. 



3. One variable quantity is said to be a "function" of 

 another, when, other things remaining the same, if the 

 value of the latter be fixed, that of the former becomes 

 determinate. This definition is due to Dirichlet ; all that 

 it implies is that for each value of the independent 

 variable there is one, and only one, definite (and 

 therefore finite) value of the dependent variable, or 

 function. 



The definition of a " continuous function " is a matter 

 of some nicety, the difficulty being apparently to frame 

 a definition which shall not merely be sufficient, but shall 

 embody a test which can be immediately applied to any 

 proposed analytical function. The definition now usually 

 adopted is as follows : 



Let x and y be corresponding values of the inde- 

 pendent variable and of the function. Let Sx be any 

 admissible increment of x, and Sy the corresponding 



