507 



D denoting ?vs usual the operation 1 + A. In order to give in suc- 

 cession the numbers u x , u x _ v u x _ 2 , . . . the machine would simply 

 have to be set to 



>f D'WJ, be used to denote u x _ lt and A' to denote D' 1. But 



D'=D~ ! , and A'=D'-l=D-'-l = D-'A, so that the required 

 numbers are 



w, -A 3 D~X A'D-V,, 

 or 



u x -Aw, A% r _, -A 3 M,-i AX-s- 



Hence the numbers on the top, A 9 , and A 4 tiers are the same as for 

 the forward calculation, while those on the A and A 3 tiers are the 

 arithmetical complements of the numbers found on those tiers after 

 the machine has made one complete movement in calculating for- 

 wards from u x . The printing part, however, is not adapted to 

 such a change : the numbers would be printed off correctly, but in a 

 wrong order ; so that unless some reversing movement were intro- 

 duced into the printing part, the printed results would only serve to 

 set types from. 



In the example chosen above, and in similar cases, the differences 

 required for setting the machine would be calculated from their 

 mathematical expressions. It might, however, be required to tabu- 

 late for small intervals a function which had been given by observa- 

 tion for larger ones, or to tabulate a mathematical function of so 

 complicated a form that the differences could not be got directly 

 without great trouble. In such a case there would be no difficulty ; 

 the differences for the smaller intervals would first have to be calcu- 

 lated from those for the larger ones by formulae in finite differences, 

 and then the setting and working of the machine would proceed as 

 before. 



It must be confessed, however, that except in the case of mathe- 

 matical tables like those of sines, cosines, logarithms, &c., it is not 

 ordinarily required to tabulate functions to intervals at all approach- 

 ing, in closeness, to those in the example selected. Hence it is 

 mainly, as it seems to us, in the computation of mathematical tables 

 that the machine of M. Scheutz would come into use. The most 



VOL. VII. 2 ' 



