SECTION III., 1882. RS] Trans. Roy. Soc. CANADA. 
On the Law of Facility of Error in the sum of n Independent Quantities, each Accurate 
to the Nearest Degree. By CHARLES CARPMAEL, M.A. 
(Read May 26, 1882.) 
ILES ENTRER a, be the » quantities true to the nearest unit, and let the abso- 
lutiesvaluerthe Git a5 Oy We, Gps osee es DIT 0) HOE nts Ws E eee T,, are each 
between the limits —4 and + 1, and all values between these limits are equally likely. 
The chance of an error in a between x, and x, + Ox, in a between x and 2, + 6m, &e., 
occurring simultaneously is 
Ohm, Ollien (lipovsectocone OL, 
: n . . 
and the chance that the sum of the errors lies between —5 and any magnitude y is 
jie ne ises ee far QU RG eee cneesare Lon 
the limits for integration being for each variable —} and 4, with the further condition that 
the algebraic sum of the variables shall not exceed y. For this integral with these limits 
I adopt the following notation, 
H LKR 
he i} |) dE i eee UT AE 7/1] 
SEL 
with a similar notation in other cases. 
To each of the variables add one-half, the integral becomes 
a settee ceeeey : AX; AL 275... A2, | ae ae y] : 
SOON OMe 0 0 2 (i) 
To find the chance of the error of the sum lying between — Z and 7, we have therefore 
to evaluate the integral (i). 
To do this let us consider the integral 
J À A Gi, As Aa eee - dr, [Er<£ +y] (ii) 
i E 1 Z. n 
the value of which is well known, and is is Ce y). 
Let ¢ be the greatest integer ins GE Then if r be any number not greater than #, 
there will be a portion of (ii) in which r chosen variables and these > only are greater than 
unity. Let us call this part R. 
Sec. III. 1882. 2 
