464 
MATHEMATICS: H. S. WHITE 
tubular necrosis and interstitial edema. Again, atoxyl causes extensive 
hemorrhage extending from the boundary zone through the medulla 
and only slightly into the cortex, the outer rim of which remains pale; 
there is marked degeneration and necrosis of tubular epithelium. These 
atoxyl kidneys which are outwardly pale combine in a peculiar form the 
essential features of both the red and the pale kidneys. In like manner, 
arsenophenylglycine produces kidney lesions of a combined type and 
while tubular degeneration and necrosis are dominant there is usually 
some congestion and hemorrhage in the boundary zone and medulla — 
more rarely in the cortex. 
The other extreme in the action of arsenicals upon the kidney is ex- 
emplified by arsacetin which produces a typically pale kidney. While 
congestion and hemorrhage are still apparent to a minor degree in the 
boundary zone of these kidneys the vascular injury is so completely 
overshadov/ed by the injury to the tubular epithelium as to leave no 
doubt as to the dominance of tubular injury. Further, the prompt 
and vigorous regeneration of the tubular epithelium indicates that the 
extensive necrosis produced by arsacetin can not be regarded as a 
secondary anemic phenomenon. 
It is certain, therefore, that all arsenicals do not produce renal lesions 
that are identical either in character or distribution but that this group 
of substances includes agents producing a so-called tubular nephritis 
as well as those producing a vascular nephritis, and that these wide 
differences in the pathogenic action of different compounds of arsenic 
are explainable only upon the basis of their chemical constitution. 
SEVEN POINTS ON A TWISTED CUBIC CURVE 
By H. S. White 
DEPARTMENT OF MATHEMATICS. VASSAR COLLEGE 
Preseented to the Academy, Aaguit 2, 1915 
Six points in space, barring special situations, determine a twisted 
cubic curve. From any seventh point of the curve those six are pro- 
jected by six generators of a quadric cone. For any seven points of a 
cubic curve there is accordingly a symmetric set of seven cones; and it 
is well known that seven points giving rise to two such cones are on a 
cubic curve, and so give rise to five more cones. This is the only current 
theorem on seven points of a twisted cubic. Concerning eight points 
there is the elegant theorem of von Staudt, that if two tetraedrons have 
eight points of a twisted cubic for vertices, their eight faces osculate 
a second cubic curve. I propose to demonstrate a theorem whose 
formulation resembles the latter, while like the former it relates to the 
