ME. W. CROOKES OX THE ATOMIC WEIGHT OF THALLIUM. 
O o ^ 
o2 i 
I wish it to be noted that I have made determinations with considerable weights of 
thallium. In ordinary analysis chemists are satisfied to take 5 or 10 grains of the 
substance under investigation : here I have gone to the very highest weight that can 
he entrusted, with safety, to the balance. The lowest weight of thallium taken is 
183-790232 grains, the heaviest 497-972995 grains, the remaining determinations 
varying between these limits. It is hardly necessary to say that the purpose has been 
to eliminate the error arising from manipulation with small quantities, and to produce 
such variety in the results as to render the chances of coincidence of very small 
value. 
Let me now tabulate the results of the determination, with the view to ascertain 
severally their degree of approximation to the arithmetic mean : — 
A. 
203-666 
+ •024 
B. 
203-628 
-•014 
C. 
203-632 
-•010 
D. 
203-649 
+ •007 
E. 
203-642 
•000 
F. 
203-636 
-•006 
G. 
203-639 
-•003 
IT. 
203-650 
+ -00S 
I. 
203-644 
+ •002 
K. 
203-638 
-•004 
The arithmetic mean of the ten observations is 
2036-424 
10 
=203-642. 
But does this average represent the truth l Or, rather, how nearly does it represent 
the truth X 
According to the theory of probabilities, the “weight” of k may be determined by 
means of the formula 
rr 
W== 22e 2 ’ 
where n= the number of observations, and 2c 2 = the sum of the squares of the successive 
differences, obtained by subtracting each observation from the arithmetic mean of the 
ivliole. 
The arithmetic mean of the ten weighings is cs=:203-042. 
2x2 
