240 NEW MODIi OF REPRESENTING 



If S and A in the calculus of finite differences, 



correspond with and d in the differential calculus ; 



the difficulties which will have to be overcome to 

 establish such a calculus as the one above men- 

 tioned, may be shown to some extent by the 

 following considerations : 



Suppose, for instance, j/ = S -.f(.t') A <r, where 



f (cT) is some unknown function of x, which must 

 be determined before the value of^can be obtained. 

 But if the object of the inquiry were to determine 

 the unknown function of x, such that the value of 

 y must satisfy one or two conditions given by the 

 particular nature of the inquiry — It would then be 

 the object of such a calculus of the variations of 

 finite differences, which I have ventured to sug- 

 gest, to determine the unknown function of x. 

 The principles of this calculus have not, to the 

 best of my knowledge, been given, nor has the 

 existence of such a calculus ever been pointed 

 out, or a particular question been proposed, 

 requiring its assistance, before the one above 

 given. 



(7). By referring to Fig. 2 ; and putting 

 AB =: o, AB' - 6, AB" = c, &c. &c. And take 



