MATHEMATICS: G. M, GREEN 209 
Eleven Cepheids with Variable Spectra 
STAR 
MAXIMUM 
MAGNITUDE 
RANGE OF 
VARIATION 
PERIOD 
NUMBER OF 
SPECTROGRAMS 
OBSERVED 
SPECTRUM VARI 
ATION 
mag. 
days 
TU 
Cassiopeiae.. . . 
7.3 
1.1 
2.139 
9 
FO to F6 
SU 
Cassiopeiae.. . . 
5.9 
0.4 
1.950 
19 
A8 to F5 
sz 
Tauri 
7.2 
0.5 
3 . 148 
11 
F4 to G2 
T 
Monocerotis. . . 
6.0 
0.8 
27.012 
6 
F4 to F8 
RT 
5.0 
0.9 
3.728 
12 
A8 to GO 
W 
Geminorum . . . 
6.4 
1.3 
7.916 
10 
F3 to GO 
RS 
Bootis 
8.9 
1.1 
0.377 
13 
B8 to FO 
X 
Sagittarii 
4.4 
0.6 
7.012 
5 
F2 to G 
Y 
Ophiuchi 
6.2 
0.8 
17.121 
4 
F5 to GO 
RR 
Lyrac 
6.8 
0.9 
0.567 
17 
B9 to F2 
6 
Cephei 
3.5 
0.8 
5.366 
21 
F2 to G3 
smaller section of the spectrum has been shown in an earlier communi- 
cation by Adams and Shapley. 
Every variable for which the present test is sufficient was found to 
vary in spectrum. It appears safe to infer, therefore, that all Cepheids 
(including the cluster- type) , besides being variable in hght and in ve- 
locity, vary periodically in spectral class as well. 
1 Shapley and Shapley, these Proceedings, 1, 452 (1915); Shapley, Ibid., 2, 132 (1916) 
Adams and Shapley, Ibid., 2, 136 (1916). 
« These Proceedings, 2, 143 (1916). 
ON THE LINEAR DEPENDENCE OF FUNCTIONS OF SEVERAL 
VARIABLES, AND CERTAIN COMPLETELY INTEGRABLE 
SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 
By Gabriel M. Green 
DEPARTMENT OF MATHEMATICS. HARVARD UNIVERSITY 
Received by the Academy, February 25, 1916 
The study of an ordinary homogeneous linear differential equation 
of the nth order leads very naturally to the definition of the Wronskian 
of n solutions of the equation, and thence to the general theory of the 
linear dependence of n functions of a single variable. This is due to 
the characteristic property of the said differential equation, viz., that 
any solution of the equation is linearly dependent upon any funda- 
mental set of solutions. I wish in this note to give some of the results 
which I have obtained in generalizing the theory of linear dependence 
to the case of n functions of several independent variables, and also to 
point out the appHcation of these results to the study of an important 
