118 THEORY OF COLLINEATIONS. 
istransformed. Along any other line through O as OS making 
an angle @ with / we have 
1 uf 
1 5 
0S OS ~ os, — sind. 
Let A, any point on /, be the vertex of an invariant pencil of 
rays and let AO=d. The elation S’ transforms AP into 
AP,and AP, into AK perpendicular tol at A. Let the angle 
PAO=}4, P,AO=>9,, P-AO=9,, ete. Along OP we have 
1 1 1 
OP = OP = OPIe 
But ae > = =. 7 =t. Substituting these 
values we have 
cot, — cote =dt. (C1) 
Thus we have the expression for a one-dimensional para- 
bolic transformation of the pencil of lines through a point 
on l, 
THEOREM 29. Anelation is completely determined by its funda- 
mental invariant figure and a single characteristic constant. The 
parabolic constant of the one-dimensional transformation along 
any given invariant line // of the fundamental figure is ¢ sin @, where 
tis the parabolic constant along the line perpendicular to the axis 
and @ is the angle which /’ makes with the axis. The parabolic con- 
stant of the one-dimensional transformation in any given invariant 
pencil of the fundamental figure is 7¢, where d is the distance along 
the axis from the vertex 0 of the elation to the vertex of the given 
pencil. 
146. Type V a Special Case of Type II and of Type III. 
We showed in the last article how type IV might be consid- 
ered as a special case of type II, whent=0Ointype II. We 
shall now show that type V is also a special case of type II. 
In type II when k, the characteristic cross-ratio of the trans- 
formation along AB and through A, is unity, these two 
one-dimensional transformations are both identical transfor- 
mations and hence every ray through A is an invariant ray; 
therefore for k = 7, the transformation T’(kt) of type II de- 
generates into S’ of type V. 
