516 
MATHEMATICS: G. M. GREEN 
this type. KHSO4 affords a notable example where three curves of 
this t3^e meet in a triple point. For a liquid, Av always decreases 
with increasing temperature on either a rising or a falling curve. On 
the rising transition curves there are 37 cases of normal variation of 
Av and 5 of abnormal variation; on the falling curves 8 normal and 8 
abnormal cases. 
The relative compressibility, thermal expansion, and specific heat 
of neighboring phases is significant. It is natural to expect that the 
phase of smaller volume will have the smaller compressibility and thermal 
expansion, and that the phase stable at the higher temperature will 
have the higher specific heat. If we call this behavior 'normal,' then 
on rising curves we find 9 cases of normal and 1 1 of abnormal compressi- 
bility, and on falling curves 1 normal and 7 abnormal. The expansion 
shows 5 normal and 7 abnormal cases on rising curves and 2 normal 
and 4 abnormal on falling curves. Cp is normal in 5 cases and abnormal 
in 7 cases on rising curves, and normal in 6 cases and abnormal in 1 on 
falling curves. The fact of abnormal Cp is of considerable significance 
from the point of view of the quantum hypothesis. It means (if we may 
apply the same considerations to Cp as to which is usually done) 
that the specific heat curves of the two modifications cannot be of the 
same character, but that somewhere between the transition point and 
absolute zero the one which is lower at the transition point must cross 
and lie above the other. 
In addition to the substances enumerated above, about 100 others 
have been examined without finding other forms. 
^ P. W. Bridgman, Proc. Amer. Acad., 47,439-558 (1912); Physic. Rev., 3, 126-203 (1914). 
ON ISOTHERMALLY CONJUGATE NETS OF SPACE CURVES 
By Gabriel M. Green 
DEPARTMENT OF MATHEMATICS. HARVARD UNIVERSITY 
Presented to the Academy. August 10. 1915 
Bianchi^ has called a parametric net of curves on a surface isother- 
mally conjugate if, when the surface is referred to these curves, the 
second fundamental form, D du^ + 2D' du dv + dv^, may by a trans- 
formation u = U (u),v = V (v) he made to take on the same shape 
as does the first fundamental form when the parametric net is iso- 
thermal; i.e., the parametric net is isothermally conjugate if D' = 0, 
D = These nets have lately attained increased importance, so 
that Wilczynski's recent geometric interpretation^ of Bianchi's condi- 
