160 PROCEEDINGS OF THE AMERICAN ACADEMY 



the readings below A should be treated in the same way as those 

 above that point, and this case will therefore not be further consid- 

 ered. Select any one of these as the second point of which the error 

 is to be arbitrarily assumed as zero, and call this B. Then 



A -[. u' -\- u" -{- + «""^ = B. 



There are thus n spaces of equal volume between A and B, and 



these correspond each to -th of the interval B — A. Hence the true 



'■ n 



reading (which, however, it is not necessary to compute numerically) 

 at the point 



And the error, obtained by subtracting the true readings as given in 

 the right-hand column from the corresponding actual reading given 

 in the left-hand column, at 



A is 



AJ^u' " A-\-u'-{A + l{B-A)}=u'-l{B-A) ^ 



A-\-u'-\-u" " u' -]- u" — ^ (B — A) 



B "0 



A — w' " —w'-\-^(B— A) 



A — w' — w" " —w' — w" -\-- (B — A) 

 &c. &c. 



In selecting B it might have been assumed equal to ^ -|~ "'> thus 

 makino-n = l. This would somewhat simplify the calculation, and 

 would be of equal accuracy, but is objectionable from the fact that in 

 general this volume would differ considerably from the average volume 

 obtained when n has a greater value (always an integer), and the re- 

 sulting series of errors would assume larger numerical values. 



