with special reference to solutions of its salts. 
565 
Let I Q ' and V be the approximate values ; 7/ + dl 0 and X' + dX 
the most probable values, so that 7 0 = Id + 77 0 and X = X' + dX. 
Then, if 7 is the observed activity of a layer of thickness x, 
7 = 7 0 (1 —e~ Kx ) = (Id + did) (1 - e - (*'+«*)*). 
Expanding by Taylor’s theorem and neglecting terms in (dX ) 2 and 
dI 0 clX, we have 
cll 0 (1 - e~ K ' x ) + xIde~ K ' x . dX = I — I' (5), 
where 7' = 7/ (1 — e~ K ' x ), the value of 7 calculated from the approxi- 
mate values of Id and X'. 
Substituting in (5) the observed values of 7 and x, and the 
approximate values Id and X', we obtain a series of linear equations 
in cll 0 and dX. These equations are solved by the usual least 
square method. 
The following results were obtained. 
For the solution of Potassium Carbonate 
\ = 12 54 ±0*57, .-. X/p = 9-016 ± 0T1, 
7 0 = 78-34 ± 1-45 and 7 0 . X/p = 706 ± 18. 
For the solid Potassium Sulphate 
X/p = 8-23 ±0-1, 
7 (l = 204-2 ± 2-9 and 7 0 . X/p = 1680 ± 37. 
The solution contains 37'5 x ffg °/ Q of KX) ; the solid contains 
t 9 7 4 ? °/o KoO. Hence the calculated values of 7 0 /a . X/p are 
2766 + 70 from the solution, 
3107 i 68 from the solid. 
§ 10. There is not perfect agreement between these numbers, 
but the interesting result appears that the value of X/p is actually 
greater for the solution than for the solid and, therefore, that the 
discrepancy introduced by the difference in these values is of the 
order and in the direction necessary to explain the apparent 
activity of thick layers. Indeed it is not to be expected that 
the value of X obtained by the measurements which have just 
been described should be such as to give a constant value of I 0 /a.X/p. 
For it has been shown previously that the rays from potassium are 
not homogeneous, but differ considerably in penetrating powers. 
The absorption does not follow an exponential law strictly and the 
value of X obtained is only a mean value of the many different 
values. This conclusion is confirmed by the present measurements, 
for the probable error of 7 0 deduced is much greater, especially in 
the case of the solid, than that which would have been expected 
from the known accuracy of the observations. The calculated 
probable error is too great because the relation between the 
intensity and the thickness is not really indicated accurately by 
equation (4) in which X is taken as the same for all rays. 
37—2 
