Vol.. 7, 1921 MATHEMATICS: E. B. STOUFFER 
273 
The calculation of the L and M, etc. critical absorption frequencies 
presents very great difficulties, for, if we suppose that an electron is re- 
moved 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 cal- 
culations. 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 absorp- 
tion wave-lengths in the L series, and six critical absorption wave-lengths 
in the M series, etc. 
According to Sommerfeld's theory 3 the difference between two L ab- 
sorption frequencies is due to the difference in shape of a circular and an 
elliptic orbit. His formula contains an undetermined constant. Pro- 
fessor Patterson and I have shown 4 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 L\ 
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. 
3 Atombau and Spektrallinien, Chapter 5. 
4 These Proceedings, Sept., 1920, p. 517. 
SEMI-COVARIANTS OF A GENERAL SYSTEM OF LINEAR 
HOMOGENEOUS DIFFERENTIAL EQUATIONS 
By E. B. Stouffer 
Department oe Mathematics, University of Kansas 
Communicated by E. J. Wilczynski, Aug. 13, 1921 
It is known 1 that the most general transformation of the dependent 
variables which converts the system of linear homogeneous differential 
equations 
m — 1 n 
yl m) + 2 2 ( 7 ) pm y * = °- ( *" = 2 - n) ' {A) 
1 = 0 k=l 
