544 Royal Society : — 



The first term in this expression will usually be the most important ; 

 and for practical purposes the expression may be still further simpli- 

 fied. If « be tolerably large, the product n . n— 1 . n— 2 . n — 3 . n — 4 

 may be replaced without material error by the fifth power of the 

 arithmetic mean of the factors, or by (« — 2)\ Again, if y be the 

 variable of which m is a function, x being merely the numeral 

 marking the number of increments of y, each equal to k, we shall 

 have near enough 



so that 



120^ ' dy" 

 In expressing a number to eight decimal places, we are always 

 liable to an error which may amount to 5 in the ninth place. 

 Hence 10"^ X 5 may be regarded as the greatest allowable error, 

 though in truth the error should not be allowed to amount to this, 

 if we wish to have the last figure true to the nearest decimal. 

 Equating then E to 10"^ X 5, we find 



„ , -0000006 \* 1 ,„. 



«=2+ — -P-— .^. (3) 



/ -0000006 y 



which gives the greatest number n of times the machine may be 

 worked without stopping and fresh setting, so far as the limitation 

 depends on the cause of error now under consideration. The in- 

 crement of y during the action of the machine, which is equal to nk, 

 or to (n—2)k nearly, n being large compared with 2, is therefore 

 nearly independent of the closeness or wideness of the intervals for 

 which the value of the function is required, a given range, so to 

 speak, of the function being taken in. Hence, so far as this cause 

 of limitation is concerned, the utility of the machine will be propor- 

 tional to the closeness of the intervals for which it is desired to tabu- 

 late the function. 



Let us now consider the effect of the decimals omitted, retaining 

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

 and higher orders has been already investigated. Let E,, Ej, Eg, E^ 

 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 Am^, ^"u^, A^u^^ 



A*u^, although they are really the errors in Am^,_,, &c., and we 



shall have for the error (E) in m^^^ 



„ n^ ,«.« — 1„ , M.w — l.w— 2„ , n.n—l.n — 2.n—3 -r, 



ii=— E,H E„+ E, + ^ E,: 



1 1 1.2 - 1.2.3 ^ 1.2.3.4 * 



or, replacing the products as before, 



£=«E,-f-i(«-l)X+l(„-l)sE3+^^(«-|)X. . . (4) 



