CONrEMPORARY ADVANCES IN PHYSICS, XXIX 295 



number j and the magnitude ViO' + 1); for the sodium atom j may 

 have the various values 1/2, 3/2, 5/2 ■ ■ • . 



I next give the rule for compounding the last of these three angular 

 momenta out of the first two, it being as I earlier said a "sort of re- 

 sultant" thereof. 



Rule for compounding I and s into j. If I is greater than .t, start 

 with the numerical value of (/ + s) and write down all the sequence of 

 numbers spaced at unit intervals from (/ + s) to (/ — s) inclusive: 

 to wit (/ + 5), (/ + 5 - 1), {l-\- s - 2), ■•■ (l - s). These (2^ + 1) 

 numbers are the permitted values of the quantum-number j. If on 

 the other hand 5 is greater than /, start with the numerical value of 

 (s -f /) and write down all the sequence of numbers spaced at unit 

 intervals from (s + /) to (s — I) inclusive: to wit, (s -{- 1) , {s -\- I — 1) , 

 {s -\- I — 2) • • • {s — I). These (2/ — 1) numbers are the permitted 

 values of the quantum-number j. 



(It will be noticed that this rule is much more generally phrased 

 than is required for the case of sodium, where 5 = 1/2, and it suffices 

 to say that j = 1/2 for / = and j = / ± 1/2 for / > 0. If, however, 

 we were dealing with an atom having more than one valence-electron, 

 5 might be replaced by a quantum-number different from 1/2 — ^not 

 because the individual electrons would have new values of spin, but 

 because the spins of two or more of them would be compounded — and 

 the general phrasing of the rule would then be required). And now, 

 to close (temporarily) the sequence of quantum-numbers and of rules: 



Suppose the atom immersed in a magnetic field of strength H, 

 parallel to the s-direction. It may then take any of several distinct 

 permitted orientations, these being denoted by various values of a 

 quantum-number m j. In any such orientation the projection of the 

 angular momentum of the atom-as-a-whole upon the z-direction or 

 field-direction is equal to mj{hl2ir). To ascertain how many per- 

 mitted orientations there are and what are the corresponding projec- 

 tions, start with the numerical value of j and write down all the 

 sequence of numbers spaced at unit intervals from + j to — j, inclu- 

 sive: to wit, j, j — 1,7 — 2, • • • — j. These {2j -f- 1) numbers are the 

 permitted values of the quantum-number Wy. 



We now start out upon a train of reasoning which leads to the 

 remarkable verifiable formula already once alluded to, the successful- 

 ness of which speaks more powerfully than any other single test for the 

 rightness of this elaborate hypothetical structure which has been 

 devised for the atom. 



Note, in the first place, that the rule given above for the permitted 

 orientations of the atom in the applied magnetic field is in accordance 



