232 BELL SYSTEM TECHNICAL JOURNAL 



tained with minerals and glasses containing these elements, and most 

 of them agree well with one or another of the theoretical curves; in 

 which connection there is an interesting detail, which I will bring up at 

 the end of the article. One would scarcely expect a theory worked out 

 for gases to apply so well to solids, and as a matter of fact it is a pe- 

 culiarity of the rare-earth atoms that even when incorporated in a 

 compound or a solid they behave in several ways more like the atoms of 

 a gas, than happens with any other elements. 



Pray do not think, however, that all this time I have been talking 

 about a theory which has no application excepting to the rarest of all 

 elements under the rarest of all temperatures. Its applications are a 

 good deal wider than that. True it is that with gases universally, and 

 with other substances ordinarily, we cannot get data along the curvy 

 parts of the curves; but we can make measurements along the sensibly- 

 linear parts near the origin. This amounts to saying that we can de- 

 termine the slope of the curve at the origin. Now of course it sounds 

 ridiculous to speak of confirming a theoretical curve by measuring its 

 tangent at one point. In this case, however, it is not altogether 

 ridiculous. Usually the experiments are made by varying H and 

 measuring / while the temperature is kept constant. Suppose this Is 

 done for several different temperatures, and suppose the results are 

 plotted by using H instead of fxHjkT for the abscissa. Then the theory 

 supplies us with different curves for the different temperatures, all 

 having the same general aspect, but different slopes at the origin. I 

 will denote these slopes for the time being by tan do. The theory, then, 

 requires that tan do should be proportional to I IT; and for gases, this is 

 found to be the case. Of course this is not such good evidence for the 

 theory as would be a complete following-up of the curve nearly all the 

 way to the asymptote; but it is pretty good by itself, and for further 

 evidence we can invoke those experimental curves for gadolinium sul- 

 phate and other solids of which I just spoke. 



If now we let ourselves be convinced by this evidence, a valuable 

 conclusion follows. From the slopes of these curves at the origin, the 

 value of n can be deduced. Let us go back to the curve of / versus 

 fj-HlkT or a, which is the epitome of all the rest. We write: 



dl/da = N,x(l - tanh^a), (5) 



(dlldll) r=const. = (dllda) (dajdll) = ( 1 - tanh^ a) Nfx ■ (iilkT) . (6) 



Since measurements are actually made at a fixed temperature and refer 

 to the slope of the curve near zero field strength, we evaluate this 

 derivative for a = 0, and we get: 



tan do = {dIldH)„^ o= Nix'^jhT, (7) 



