432 Wisconsin Academy of Sciences, AHs, and Letters. ■ 



§ 3. 



Tlius the error of computation takes the form' 



in which the coefficients « are known, but in which only the lim- 

 its of the quantities /i, /g? ^^c, are determined. 



Assume each / to have this limiting value, and add the prod- 

 ucts 



the signs + or — being disregarded, and the result is the great- 

 est possible error 



= ^ («/); 

 the extreme limits within which the error of the computation 

 must necessarily be included are 



— '2 (af) and + 2 {af). 



But the error will never reach these extreme limits, since each 

 error included within the expression frequently diminishes to 

 an infinitesimal, so that the maximum error as given above is 

 not a suitable test by which to measure the accuracy of the com- 

 putation. If for each error lying within these limits a propor- 

 tional number could be determined, which would exhibit the 

 relation that holds between the number of errors of that mag- 

 nitude ' and the total number of errors of every magnitude, 

 or which shows how many errors of a certain magnitude there 

 would be among all the errors possible, we should have enough 

 proportionals to test the accuracy of any formula. 



Such numbers may be found if first we give to each f all the 

 different values that are possible, then substitute in the sum 



all the combinations of the different quantities ; in this way 

 we may observe how often the sum will equal zero, and how 

 often it will be equal to any other given number within its ex- 

 treme limits. 



