ISO BELL SYSTEM TECHNICAL JOURNAL 



function of height, over an interval of this magnitude. This makes 

 the problem of determining the trend of that curve (if such a curve 

 there be) a formidable one. 



Since the problem is so formidable and so important, let us linger 

 over it. It is expedient to treat the very simplest conceivable case, a 

 case which is certainly much simpler than reality, yet quite complicated 

 enough for a first view. Suppose a beam of photons of a single wave- 

 length descending vertically, first through vacuum, then (at x = 0) 

 entering a horizontal sheet of some medium (water, say) of uniform 

 density, in which they have an absorption-coefiicient /z. Assume that 

 every impact of a photon against an electron results in the total 

 disappearance of the former, in the projection of the latter straight 

 onward (vertically downward). Assume further that the projected 

 electron engenders a constant number of ion-pairs per centimetre of 

 its path, and that the length R of the path is the same for all the 

 electrons. 



Then, one readily sees that the ionization in the water, instead of 

 being greatest at the top and diminishing steadily downward, actually 

 increases from the top down to the depth R, and begins to decrease 

 only beyond R. The formula is as follows, x standing for distance 

 measured from the surface of the water downwards: 



J cc 1 - e-""" for X < R, 



I o: c-'^ie"" - 1) for X > R. 



There is a "zone of transition" of thickness R, beyond which the 

 ionization diminishes exponentially, w^ith the same exponent as we 

 have assumed for the absorption of the gamma-rays themselves. This 

 is the exceptional case mentioned above, in which by diving to depths 

 exceeding R one could arrive at a region where the trend of the ioniza- 

 tion is the same as that of the strength of the beam of photons, and 

 the value found for ^ would apply to these last. 



Unfortunately there is no actual case so simple. If there are cosmic 

 gamma-rays coming from the sky, they evidently come from all 

 directions, not merely vertically; and from the character of Millikan's 

 curves it seems likely that they are of several or many wave-lengths, 

 not one only. Conceivably at great depths of water the hardest of 

 all may be filtered out, and these depths may be superior to the values 

 of R for all the electrons; if so then the lowest values of /x, quoted by 

 Millikan and by Regener, may pertain to actual photons. If there are 

 cosmic gamma-rays of several frequencies descending vertically from 

 above, then the actual ionization-curve should be a sum of terms such 

 as (3), with numerical factors depending on the frequencies and the 



