270 S. Newcomb and EF. S. Holden on the Periodic 
Our results should, however, depend, not on a simple enumera- 
tion and comparison of the signs of the residuals, but also on 
the magnitude of the latter, and we may secure this dependence 
by taking the algebraic product of each residual of the one 
series by the corresponding one of the other. If the residuals 
are purely accidental, the mean value of these products should 
approximate to zero as the number of observations is increased, 
while in the case of actual variability it will approximate to 
Aas limit. Let us investigate exactly what this limit 
will be. 
If we have two determinations of any quantity, each affected 
y @ common but unknown error s, and also by independent 
accidental errors r and r’, whose law of probability is that 
tions are made; it is required to find the mean value of the 
product (s + r) (s + 7’). 
If the measure of precision of the determinations is put 
equal to unity, the probability that any error of one observa- 
tion of a pair will fall between the limits s+r and s+r+dr is 
ee _oe — "dr ; 
the probability that the error of the other observation of the 
pair will fall between the limits s+7’ and s+7r’+dr"’ is 
See ae 
7 
The probability of the combination is therefore 
lo rp y's’ 
—.e .e 
. ar. dr’, 
This probability multiplied by the product of the errors is 
1 ~- - 
oan (s+r)e (oe rye f. ov dr’. 
The mean value of the product required is the sum of all 
these products, as r and r’ each varies independently from + ® 
to — om, or the double integral 
e 1 cies ose t 
=f (str)e~ "Tepes! dr. dr’. 
-0  -o 
Integrating first with respect to r’ we find 
+ CO a +00 
# & (¢+r') e~ dal oH" ay mae 
~ co age 
eat Te ee i. 
 ?’)34 
