505 



only four orders of differences, since the effect of omitting the fifth 

 and higher orders has been already investigated. Let Ej, E 2 , E 3 , E 4 

 be the errors left in the first, second, third, and fourth differences in 

 setting the machine. Then in the same manner as before these may 

 without sensible error be regarded as the errors in Ay^, A'w^,, A 3 w a ,, 

 A 4 M^,, although they are really the errors in A^_,, &c., and we 

 shall have for the error (E) in u x+n 



n.n \ .n 2 n , n.n 1 .w '2.n 3 



2 1.2.3 1.2.3.4 



or, replacing the products as before, 



If each of the quantities E,, E 2 , E 3 , E 4 be liable to be as great as 

 10~ l6 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~ 9 x5, the greatest allowable error in E, we find 



n -=(24x10-)*. = 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 4 M^, is smaller, perhaps much smaller, than 10~ 1<5 x5, 

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

 A 4 w^ or the possible value 10~ l6 x5 of E 3 , as the case may be. 

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

 the third term in the second member of (4) to 10~ 9 x5, =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 



M=siny, k= - - - = '0002909, ^. = cosy. 

 180x60 dy* 



Putting for this last differential coefficient its greatest value unity, 

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

 ence is very nearly equal to k 4 s'my, which may contain figures in 

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

 value of n in consequence of the restriction arising from decimals 



