503 



being employed in the whole construction except two, the office of 

 which is quite subordinate. When the parts are moved, they remain 

 in their new places either from their weight or from friction, there 

 being nothing to disturb them. This circumstance, which renders 

 a wilful derangement of the machine exceedingly easy, permits of 

 great simplicity and consequent cheapness of construction ; nor does 

 the machine seem likely to get out of order if reasonable care be 

 taken of it. 



The machine is competent to tabulate to any extent a function 

 whose fourth differences are constant, so long as the expression of 

 the numerical value of the function does not involve more than eight 

 digits. The most general form of such a function is of course 



a + bx + cx* + dx 3 (1) 



Were the machine restricted to such functions, its use would be 

 limited indeed ; its utility must of course depend on its being ap- 

 plicable to functions in general, which, except in singular cases, may 

 be expressed within a limited range of values of the variable # by a 

 function of the above form. To estimate the capacities of the ma- 

 chine, or rather of a difference engine in general, whatever may be 

 its particular cpnstruction, it will be necessary to investigate how 

 soon the quantities neglected begin to tell in the result. 



Now these quantities are of two kinds ; first, the fifth and higher 

 differences; secondly, the decimals of the fifteenth place. The effect 

 of these may be examined separately. We may always suppose the 

 first spindle to represent the first place of decimals, since it will 

 only be necessary to multiply or divide by some power of 10 should 

 that not be the case. 



Suppose the machine set for u x , and its first four differences (or 

 to speak more exactly, the differences ^u x _ lt A 8 M iJ ._ 1 , A%_j,_ 2 , 

 AXs)' aQ d worked n periods, so as to give what ought to be u x+n . 

 We have 



*,+=**+ ito.+ t^^V*** (2); 



and since the machine would give u x+n exactly if the fourth differ- 

 ences were constant, the error (E) will be 



n.n l.n 2.w 3.w 4 A . . . !. 2.n 3.n 4. 5 . . 

 1.2.3.4.5 ~ A5 "*+- 1.2.3.4.5.6 *"* 



* This expression will not be absolutely exact, since it is Au^_ 1 , &' 2 u x _ l , 



