Report on the Calculating Machine of M. Scheutz. 545 



If each of the quantities Ej, Eo, Eg, E^ be liable to be as great as 

 10"' x5, the last term in this expression will be the most im- 

 portant if n be considerably greater than 4, Equating this term to 

 10~ X 5, the greatest allowable error in E, we find 



«-|= (24x107)', w= 126 nearly, 



so that the machine may be worked about 100 times without fresh 

 setting. 



In practice the limitation may be even less than this ; for it may 

 happen that A-tw^ is smaller, perhaps much smaller, than 10~'^x5, 

 in which case the limitation wiU depend upon the absolute value of 

 A*M^ or the possible value 10~'®x5 of E3, as the case maybe. 

 Should the restriction arise from the latter cause, we get by equating 

 the third term in the second member of (4) to 10~^x5, w=392 

 nearly. 



To illustrate these limitations by an example, suppose that it was 

 required to make a table of sines to every minute. In this case we 

 have 



Putting for this last differential coefficient its greatest value unity, 

 and substituting in (3), we get n=196 nearly. The fourth differ- 

 ence is very nearly equal to — Ar* siny, which may contain figures in 

 the fifteenth place, so that w=126 is about the greatest allowable 

 value of n in consequence of the restriction arising from decimals 

 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 m* would be comparable with one another, 

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

 shown that 126 x2~*, 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 ilrst 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^, 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 



