346 MATHEMATICS: G. M. GREEN 
• 
nificant. The values for platinum, given for comparison, were obtained along 
with the silicates. 
Albite and microcline have a smaller proportion of oxygen atoms than silica, 
and this should bring the theoretical value A v nearer 5.96 for them than for 
silica. Moreover, the values of A P given contain the thermal effect of expan- 
sion. Allowances made for these two facts bring these substances into a class 
with the forms of silica, and thus tend to give assurance that the silica results 
are not exceptional, although the data are too uncertain to add anything to 
the exactness of our knowledge of the variations from the theoretical curve. 
Anorthite is given as a type of substances showing above 1000° a much larger 
increase. This increase, however, is very likely in part the effect of the latent 
heat from a slight amount of melting, due to a very small amount of impurity. 
Such effects have regularly been found in other kinds of work on silicates. 
But while such melting can affect the atomic heat as determined for anorthite 
and most other substances, which crystallize readily on cooling, it can scarcely 
do the same for the other silicon compounds given here, all of which are very 
sluggish in their changes of state, so that any slight portion of them melted 
at the higher temperatures would in all probability merely cool to a glass, 
during the heat determination in the calorimeter, giving out no latent heat. 
Indeed, since the effect of impurities in causing melting increases rapidly with 
the temperature, these substances are very exceptional in their value for dem- 
onstrations at high temperatures such as that which is the subject of this 
paper. 
Summary. — The specific heats of three forms of silica and two silicates 
(alkaline feldspars) determined for temperatures up to 1300°, indicate that 
the atomic heats at constant volume for these substances increase above 
theoretical value, asymptotic to 5.96, as the heats of metals have been known 
to do, and hence that such increase is a very general phenomenon, as has 
been suspected. 
1 A fuller account will be published elsewhere. 
2 Nernst, W., Ann. Physik, Leipzig, 36, 1911, (430). 
ON CERTAIN PROJECTIVE GENERALIZATIONS OF METRIC 
THEOREMS, AND THE CURVES OF DARBOUX AND 
SEGRE 
By Gabriel M. Green 
Department of Mathematics, Harvard University 
Communicated by G. D. Birkhoff, September 18, 1918 
About a year ago appeared in these Proceedings an abstract 1 of some inves- 
tigations which I was then engaged in preparing for publication. As the 
work progressed it expanded, and I wish to devote this note to a description 
