NEWSON: PROJECTIVE TRANSFORMATIONS. 67 
must coincide with the plane ABI; and also the line | must coincide 
with the line AB. The two dimensional transformation in the 
single invariant plane is of the third kind. This invariant figure 
characterizes type V, 
In the five preceding cases if any transformation leaves invariant 
the invariant figure and one other invariant point not in an inva- 
riant plane or one other invariant plane not through an invariant 
point, the transformation is an identical one, and every point, line 
and plane in space is invariant. But the transformation is not 
necessarily identical when an extra invariant point is found on an 
invariant line or in an invariant plane. 
If an extra invariant point is taken in one of the faces of the 
invariant tetrahedron of type I, but not on an invariant line, then 
all points in that face are invariant points and consequently all 
lines through the opposite vertex are invariant lines of the trans- 
formation. ‘The resulting invariant figure is self dualistic and 
characterizes type VI. The corresponding transformation is called 
a perspective transformation. The one dimensional transforma- 
tions along the invariant lines and in the invariant pencils of rays 
and of planes are all of the first kind leaving two elements inva- 
riant. 
Asa special case of the above the vertex of the bundle of invariant 
rays may lie in the plane of the invariant points. Such a figure is 
self dualistic and characterizes type VII. All the one dimensional 
transformations involved in it are of the second kind leaving one 
element invariant. The transformations of this type are called 
Elations in space. 
It should be remarked that these two types VI and VII are the 
only types of projective transformations in space that leave all the 
points of a plane invariant. This can be shown by examining all 
the possible forms of this kind to be derived from the first five 
types. If an extra invariant point be taken in one of the invariant 
planes of type I, the result is type VI. If an extra invariant point 
be taken in the plane ABC of type II, the result is type VII. If it 
be taken on either of the planes ABI or ACI, the result is type VI. 
If such a point be taken in either of the invariant planes of type 
III, the result is type VII. If such a point be taken in the inva- 
riant plane ABI of type IV, the result is type VII. But if taken in 
the plane x of type IV, the result is type VI. If such a point be 
taken in the invariant plane of type V, the result is type VII. In 
all these cases it is understood that the extra invariant point is not 
taken on an invariant line of the invariant plane. We have thus 
