NORMAL FORMS. 109 
portional to the sines of the angles which / makes with the 
y- and w-axis respectively. In either case ¢ and c’ are not in- 
dependent. 
In homogeneous coordinates the above result may be writ- 
ten in the form 
1} OF BO | 22 Mp 2 
TSAR Bu CrA |. os Bare B: : 
Oa Al Be Ciel 9 PYr—| ar BC KB , 
[eC c'CO tA+cC| cC (CO tB+e'C 
x y ma 
Aer eatC AC. 
P41-la B OKC (16) 
leG eC 6 tC 
The determinant of the normal form (15) of type II, is 
PAC Bare, \s 
IN eB) Ce 
le ce! ~0))| 
THEOREM 26. A collineation of type II is expressible in sym- 
metric determinant form in terms of its seven natural parameters. 
138. Another Method for Type II.* In the last article we 
derived type II from the general case, type I, by letting A’ 
approach A along the side AA’ of the invariant triangle. 
The following method, which is presented so as to be applica- 
ble to type III, also is more general and includes the above 
method as a special case. A, A’ and A” being the invariant 
points, draw any continuous curve s from A through A’ and A”, 
Fig. 20. Let the curve have a definite tangent and curvature 
at A, and let the direction of the tangent and the curvature 
be continuous throughout s. Now let A’ approach A along the 
curve s, A’ remaining fixed. In the determinants of the 
normal form of type I substract as above the second row from 
the third and write A’— A=AA, B/—B=AB. Divide the 
new third row through by As, the length of the are AA’. 
*The method of this article and its application in the next article is due to Dr. 
Paul Wernicke. I am under obligations to him for valuable assistance in regard to 
the normal form of type III, as here presented. 
