302 N. L. BOWEN 
calculation of the concentration at any point in a diffusion cylinder, 
the equation™ becomes 
l-« L—(2m—*) 
2V kt 2V kt 
26=Co a2. cP dg+—?- e-®d6+etc. | I 
We ae 
—l—x —l—(2m—*) 
2V kt 2V kt 
where the term in brackets in the limits is successively x, 2m—x, 
2m+x, 4m—x, 4m-+x, etc., and where c is the volume concen- 
tration at any point at distance x from the base of the column, / is 
the thickness of the bottom layer of original uniform concentration 
Co, m is the total length of column, ¢ is the time elapsed, and k the 
constant of diffusivity. For the examples in hand the series is 
rapidly convergent. With the aid of this equation we may, then, 
calculate the concentration at various points after a certain period 
of time and for a certain value of k& (or, more simply, for a certain 
value of the product kt) and draw a curve representing the theo- 
retical distribution of concentration. Curves of this kind were 
drawn and it was found that in no case could a calculated curve 
be obtained that would coincide with the observed curve. Of the 
calculated curves a certain one was chosen and was plotted on 
each of the figures as a dotted curve. The theoretical curve 
chosen in each case was that which showed approximately the 
same concentration at the upper surface as that actually found. 
The curves therefore coincide at their upper ends, but at other 
depths wide divergence is shown between the full curve of actual 
concentration and the dotted curve of theoretical concentration. 
In all cases this divergence is of a systematic kind, the actual con- 
centration showing a smaller gradient in the diopside-rich layers 
and a larger gradient in the diopside-poor layers than the theo- 
retical concentration. Thisis shown particularlyplainly in Figure 1, 
where the diopside-rich layers have reached practical uniformity 
while the upper layers show a very strong gradient. This uniform 
: : 6 9 q 2 
«The equation is not so formidable as it appears, {| of fds being merely 
the probability integral whose value, for various values of g in the limits, can be 
looked up in the tables. 
