546 •■ f* ,}l['\Royal Society :-'^n'S^'> -** ~f> »,t<-~-,^ 



the fifth root of the greatest allowable error in w,.- If, for example, 

 it were 196, it would become 311, or 492, or 780. 



The above is a fair specimen of the application of the machine. 

 The particular function chosen is, it is true, a familiar one, which 

 has been long since tabulated, but it is not the worse fitted for 

 an example on that account. It may be seen at once how much 

 mental labour and risk of error is saved by the use of such a machine, 

 when tables have to be calculated to close intervals. The whole 

 exertion of mind is confined to calculating the function and its dif- 

 ferences at wide intervals, say for every 100th or 60th number to be 

 tabulated, and setting the machine. Even this exertion (except so 

 far as relates to the setting, which is easy,) might be reduced to one 

 half, if desired, by setting the machine to calculate backwards as 

 well as forwards. In order to give in succession the numbers 



Uj,, ?',+!> Wj+2' • • • t^^ machine has to be set to 



«r '^"j-l ^X-l ^^«r-2 ^''«r-2> 



or to 



u^ M>-'u^ A*D-'„^ i^^D-\ A*D-\. 

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

 cession the numbers u^, u^_y, m,._2. ... the machine would simply 

 have to be set to 



• u^ ^'D'-\ A'^D'-y, A'3D'-X A'^D'-V!,:;^'bBfl 



if D'», be used to denote u^_i, and A' to denote D' — 1. But 

 D'=D"', and A'=:D' — 1=D"' — 1 = ~D"'A, so that the required 

 numbers are 



... u^ — Am^ AX-1 — AX-1 AX-2- 



Hence the numbers on the top, A'^, and A"* tiers are the same as for 

 the forward calculation, while those on the A and A^ 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 ?</ The printing part, however, is not adapted to 

 such a change : the numbers would be printed ofl-' 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; 



