506 



left out, which in this example is what limits the working. Should 

 the intervals be a good deal wider than 1', as 5', it would then be 

 the omission of fifth differences that would impose the limit, for 

 the greatest allowable range on this account would be nearly the 

 same as before, or about 3, which would contain only thirty-six 

 values to be calculated. Should it happen that both causes of error 

 were about equally restrictive, it must be remembered that the cor- 

 responding errors in u x would be comparable with one another, 

 and might be added together; and in this case it may easily be 

 shown that 126 x 2~*, or 106 nearly, is somewhat inferior to the 

 greatest allowable value of n. Should eight figures not be required 

 to be retained, but seven, six, or five be sufficient, the last one, two, 

 or three of the first eight spindles might be used for calculating 

 instead of printing ; and since the greatest allowable value of n, so 

 far as depends on omission of decimals, varies nearly as the fourth 

 root of the greatest allowable error in u f , that value would be in- 

 creased in the ratio of 1 to the fourth root of 10, or 100, or 1000, 

 and from 126 would become 224, or 398, or 708. The greatest 

 allowable value of n as regards the omission of fifth differences 

 would increase in a somewhat slower ratio, since it varies nearly as 

 the fifth root of the greatest allowable error in u x - 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 

 u w w' * ne machine has to be set to 



or to 



A 4 D~ 2 r , 



