578 
MATHEMATICS: G. D. BIRKHOFF 
in the upper right, with the practical absence of analyses in the upper 
left and lower right quadrants, is very evident. 
The chemical composition of igneous rocks is so complex and the 
number of their chemical constituents is so great; the conditions of 
differentiation and solidification are so varied and so compUcated; and 
the knowledge of the application of the laws of physical chemistry to 
them is as yet so meager by reason of the lack of data, that some irregu- 
larities and apparent exceptions are to be expected. Furthermore there 
are other correlative relations which have not been dealt with here, and 
it is conceivable or indeed probable that one or more of these may under 
certain conditions, whether of composition, differentiation or solidifica- 
tion, supersede the one which is the subject of the present discussion 
without invalidating the truth of this. 
It would be premature to enter here into a discussion of the causes 
of the correlation of these four elements. Any such would necessitate 
the consideration of the other correlations which have been detected. 
It may be said, however, that even though such concomitant variations 
may be due to similar solubility relations, or some such physico- 
chemical factor, yet that such similarities are themselves presumably 
due to certain intimate relationships between the elements which are 
commonly and somewhat loosely called affinities. 
1 J. H. L. Vogt., Zs. prakt. GeoL, 1898, 326. 
2 J. F. Kemp., Ore Deposits, 34-37 (1900). 
"L, de Launav, La Science Geologique, 637, 1905; Giles Mineraux, 1, 46-51 (1913). 
4 W. F. Hillebrand., Bull. U. S. Geol. Siirv., 305, 21 (1907). 
5 H. S. Washington., Trans. Amer. Inst. Min. Eng., 1908, 749-767. 
" H. S. Washington., Prof. Paper U. S. Geol. Surv., 14, 1903. 
THEOREM CONCERNING THE SINGULAR POINTS OF 
ORDINARY LINEAR DIFFERENTIAL EQUATIONS 
By George D. Birkhoff 
DEPARTMENT OF MATHEMATICS. HARVARD UNIVERSITY 
Presented to the Academy, October 15, 1 91 5 
In earlier papers I have considered the effect of a hnear transfor- 
mation of dependent variables upon the solutions of ordinary linear dif- 
ferential equations in the vicinity of a singular point. 
This type of transformation led me to the notion of equivalence which 
is fundamentally important for the classification of singular points. 
Ordinary Hnear differential equations also preserve their form under 
an arbitrary transformation of the independent variable. I shall prove 
here that this second type has no additional significance for the purposes 
