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In order to make the computations for chemical elements other than those in table 3 it appears to be necessary to make additional assumptions as to just how the elements are built up in increasing values of the atomic numbers. There is also some difficulty about the radii of the various orbits. In general, a two quantum orbit has a much smaller radius than a three quantum orbit.

The formulas of this and the preceding note contain no undetermined constants after the distribution of the electrons has been fixed; i.e., there is nothing in the equations which must be obtained from X-ray measurements themselves. The formulas are functions of numbers and of π except in the correction term for the change of mass of an electron with its velocity. The correction term is the only place where measured quantities appear. These quantities are the electron's charge e, Planck's action constant h, and the velocity of light c, and they may be regarded as having been determined by experiments other than X-ray measurements. In spite of this fact the formulas represent the observed values to within a few per cent. This is fairly close, considering the influences that have been neglected. Among the reactions that have been neglected we may mention the forces due to electrons in orbits outside the one considered, the fact that electrons inside these orbits have been supposed concentrated in the nucleus, forces due to the magnetic fields of the revolving electrons, the influence of electrons that may be out forming bonds with other chemical elements and which may not, therefore, take part in the energy changes, etc.

The fact that the various distributions considered give results that differ from each other by only a few per cent indicates that the calculation of the K critical absorption frequency is not a sensitive method of determining the exact distribution of the electrons. Further, other distributions than those considered may furnish more accurate values. The important point is that the distribution of electrons is what may be called the chemical distribution, and that the numbers of electrons in the orbits are related to their quantum numbers according to equation


The calculation of the L and M, etc. critical absorption frequencies presents very great difficulties, for, if we suppose that an electron is removed from the second or third pair of orbits, it leaves this pair of orbits unbalanced. Just what would happen in this case is not clear, and it would require an additional assumption in order to complete the calculations. Definite general conditions of the dynamic equilibrium have not yet been found.

It may be, also, that orbits that are not circular would give better values than circular orbits. Computations of the frequencies on this basis present formidable difficulties. The fact, however, that the two quantum and three quantum orbits lie not in a plane, but in space of three dimensions may explain the appearance of three critical absorption wave-lengths in the L series, and six critical absorption wave-lengths in the M series, etc.

According to Sommerfeld's theory3 the difference between two L absorption frequencies is due to the difference in shape of a circular and an elliptic orbit. His formula contains an undetermined constant. Professor Patterson and I have shown that if we assume four electrons in the L orbit the undetermined constant is done away with, and that Sommerfeld's formula represents roughly the difference between the Li and L2 absorption frequencies. It may be that a formula calculated on the basis of three dimensional orbits would give more accurate results. I am greatly indebted to several of my assistants for carrying through many of the computations.

1 These PROCEEDINGS, Sept., 1921, p. 260.

2 Nature, March 24, 1921.

Atombau and Spektrallinien, Chapter 5.
These PROCEEDINGS, Sept., 1920, p. 517.



Communicated by E. J. Wilczynski, Aug. 13, 1921

It is known that the most general transformation of the dependent variables which converts the system of linear homogeneous differential equations

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where Pikl are functions of the independent variable x, into another system of the same form is given by the equations

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where a are arbitrary functions of x and where the determinant A of the transformation does not vanish identically.

A function of the coefficients of (A) and their derivatives and of the dependent variables and their derivatives which has the same value for (A) as for any system derived from (A) by the transformation (1) is called a semi-covariant. If a semi-covariant does not contain the dependent variables or their derivatives, it is called a seminvariant. A complete system of seminvariants of system (A) has been calculated. It is the purpose of this paper to obtain the additional semi-covariants necessary for a complete system of semi-covariants.

If equations (1) are solved for y, there results

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where A is the algebraic minor of a in A. If the coefficients in (1) are assumed to satisfy the conditions for the transformation of (A) into the semi-canonical form, the successive differentiation of (2) gives


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The most general form of (1) which leaves the semi-canonical from in the semi-canonical form is given by the equations2

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where a are arbitrary constants whose determinant D is not zero. The semi-covariants in their semi-canonical form are obtained by transforming the semi-canonical form of (A) by (5). We shall let Til denote the coefficients of the semi-canonical form of (A) which correspond to the


coefficients Pikl of (A). The effect of the transformation (5) upon (7) is given by the equations2

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π i, j, m −2 rj, l −1 (i = 1, 2,...n, l = 1, 2,....n-1), (7)

where jo = yj, it is easily verified that each of the sets of quantities r(i = 1, 2,........,n) is transformed by (5) cogrediently with y; (i=1, 2,....... n). Therefore, the determinant

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is a set of quantities which are transformed by (5) cogrediently with y; (i=1, 2,................. ,n). We therefore have n

1 additional relative

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Since the coefficients in (5) are constants, each set y") (i=1, 2,...,n) of derivatives of y; are transformed by (5) cogrediently with y; (i = 1,2,...,n). We therefore have mn n relative semi-covariants

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A comparison of (3) with the inverse of (5) and of the expressions,2 and Tikl in terms of the coefficients of (A), with (6) shows that the


semi-covariants R, S, T, may be expressed as semi-covariants of (A) simply by replacing y") by tir, Tikl by uikl, and Tikl by Vikl, where Иikl and Vik are functions of the coefficients of (A) and their derivatives which appear in the expressions for Tikl and Tikl.

If the transformation (1) and the corresponding transformations for the derivatives of y; are made infinitesimal, and the resulting system of partial differential equations for the semi-covariants is set up, it is found that there are exactly mn relative semi-covariants which are not seminvariants. We thus have the proper number of semi-covariants, but it remains to show that they are independent.

A comparison of R and S; with the corresponding semi-covariants3 for the special case of (A) where m = 2 shows R and S; to be independent. Again, the functional determinant of T1, with respect to y{") (i=1,2,...,n) for each value of 7 1, 2,..... m 1 shows that Ti, are independent, among themselves and of R, of S; and of the seminvariants. We have now proved the following theorem:


All semi-covariants are functions of seminvariants and of R, S, (i = 1 2,.........,n−1), T1, (l 0, 1,...... n-1; T 1, 2,........m-1).



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1 Wilczynski, E. J., Projective Differential Geometry of Curves and Ruled Surfaces, Teubner, Leipzig, Chap. I.

2 Stouffer, these PROCEEDINGS, 6, 1920 (645–8).

3 Stouffer, London, Proc. Math. Soc., (Ser. 2), 17, 1919 (337-52).



Communicated by L. E. Dickson, April 16, 1921

1. Comprehensive as the existent theory of differential parameters is, as related to quantics

F =

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(ao, ai, am) (dx1, dx2) (aj under arbitrary functional transformations (1)

X;= X; (V1, V2) (i = 1, 2),

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aj(x1, x2)),

developments of novelty relating to the foundations result when emphasis is placed upon the domains within which concomitants of such classes may be reducible, particularly a certain domain R(1,T,A) defined in part by certain irrational expressions in the derivatives of the arbitrary functions occurring in the transformations. For a given set of forms F all differential parameters previously known are functions in R of certain elementary invariants, which we designate as invariant elements, and their derivatives. The theory of invariant elements serves, therefore, to unify known theories and, for the various categories of parameters, gives a means of classification.

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