By Noyes A.A., Beckman A.O.
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Additional resources for A Periodic Table of the Structure of Atoms and Its Relation to Ion-Formation and Valence
All the rest of his results must be excavated by the reader from the many pages of prose. In the following we shall mention some of these results and give an impression of the novel methods Sturm used to prove them. These methods are intuitively appealing but of course not rigorous in the sense of the later 19th century. However, Bôcher  later showed that all of Sturm’s arguments could be made entirely rigorous rather easily. 2 Today we interpret this identity as saying that eigenfunctions belonging to diﬀerent eigenvalues are orthogonal to each other in a Hilbert space L2 [α, β] with the measure gdx.
Moreover we want to make sure that no sign change can appear or disappear except through the end point i = n. A suﬃcient condition is that Li (r)Ni (r) > 0. Indeed if a root appears or disappears from the interior of the interval for a certain value r0 of r, Ui+1 (r0 ) must be zero for some value of i and Ui (r0 ) and Ui+2 (r0 ) must have the same sign. However, if Li (r)Ni (r) > 0 it is clear from the equation (30) that this cannot happen. By a simple change of dependent variable one can reduce the equation (30) to the form (31) Ui+1 (r) + Gi (r)Ui (r) + Ui−1 (r) = 0.
Then, clearly L has a zero at −k11 /g11 . Furthermore, the leading term of the quadratic M is the principal minor of order 2 of A, which is also positive since A > 0. Thus, M is concave up and an easy calculation shows that M (−k11 /g11 ) < 0; M therefore has two real zeros and the interval having these zeros as endpoints must contain the zero of L. Next, the leading term of the cubic N is the principal minor of order 3 of A, again positive since A is. Labeling the zeros of M by r00 , r11 , we see that a necessary and suﬃcient condition for the zeros of N to separate those of M is that N (r00 ) > 0 and that N (r11 ) < 0.
A Periodic Table of the Structure of Atoms and Its Relation to Ion-Formation and Valence by Noyes A.A., Beckman A.O.