68 KANSAS UNIVERSITY QUARTERLY. 
exhausted the possibilities and have obtained only the two types 
Vivande Vall, 
Again, another set of special types may be found by taking an 
extra invariant point on one or more of the invariant lines of the 
first five types. 
If a transformation leave invariant a third point on one of the 
edges of the tetrahedron of type I, the one dimensional transforma- 
tion along that edge is identical and every point of the edge is 
invariant. The resulting figure characterizes type VIII. It con- 
sists of all points on the line AD, the points B and C, and all 
planes through the line BC. The two dimensional transformations 
in the planes BAD and CAD are both of the fourth kind; those in 
the other invariant planes are all of the first kind. 
If a third invariant point occur on the line BC of type II, then 
all points on that line are invariant and the resulting figure charac- 
terizes type IX. The two dimensional transformation in the plane 
ABC is of the fourth kind; the two dimensional transformations in 
the invariant planes through Al are all of the second kind. 
If an extra invariant point occur on the line BC of the invariant 
figure of type VIII, the resulting figure characterizes type X. The 
invariant figure consists of all points on two non-intersecting lines 
and of all lines joining two invariant points. The one dimensional 
transformations along the invariant lines are all of the first kind. 
The two dimensional transformations in the invariant planes are all 
perspective transformations. 
In the invariant figure of type II if an extra invariant point is 
found on the.line AB, the resulting figure characterizes type XI. 
The two dimensional transformation in the plane ABI is of the 
fourth kind; that in the plane z is of the third kind; that in every 
plane through AB is of the second kind. 
In the invariant figure of type III if an extra invariant point be 
taken on the line BC, then all points on that line are invariant and 
also all planes through the line of invariant points. The resulting 
figure, consisting of all points on a line and all planes through the 
line, characterizes type XII. The two'dimensional transformations 
in the invariant planes are all of the fifth kind. 
In the invariant figure of type 1V if an extra invariant point be 
taken on the line AB, all points on that line and also all planes 
through Al are invariant. The resulting figure characterizes type 
XIII. The two dimensional transformation in the plane ABI is of 
the fifth kind, while the two dimensional transformations in the 
planes through Al are all of the third kind. 
