82 KANSAS UNIVERSITY QUARTERLY. 



of constructing a projective transformation referred to above, we 

 see that any two conies touching a line 1 determine a projective 

 transformation of the plane. Since the number of conies 

 touching the line is oo*, the number of pairs of such conies is co'^ and 

 hence there are oo** projective transformation of the plane obtained by 

 taking any line 1 as the fixed line of the construction. The line 1 was 

 taken as the line of intersection of the two planes tt and tt', and in de- 

 veloping the construction of one projection of the plane upon the oth- 

 er the angle between the two planes was not considered. By making 

 the planes tt and tt' intersect in some other Ime as 1, we get another sys- 

 tem of transformations which must be identical with the first system. 

 If the angle between the two planes in the last position is not the 

 same as in the first position, the transformations of the two systems 

 will not be in the same order, but no new transformation will be 

 introduced. We therefore infer that there are only o:>^ projective 

 transformations in the plane. 



The group of the projective transformations of the plane will be 

 called the General Projective Group and will be designated by the 

 symbol G^. 



Tlieorcni 4. There are 00**, projective transformations of the plane ; 

 these form the General Projective Group G^ whose fundamental property 

 is that any two transformations of the ^^■roi/p are together equivalent to 

 some third transformation belonging to the sante group. 



(For Lie's analytical proof see "Cont. Gruppen," Kapitel 2, §1.) 



Every projective transformation of the plane leaves some line or 

 lines and some point or points of the plane unaltered in posi- 

 tion, or as we say, invariant. There are five types of these trans- 

 formations, distinguished according to the kind of plane figure 

 which is left invariant. (See Vol. IV, page 248 K. U. Q. and "Cont. 

 Gruppen" page 35-6). If two transformations T and T, both leave 

 any plane figure invariant, e. g. a line 1, the transformation Tg 

 which is ecpiivalent to the combination of T and T^ must also 

 necessarily leave 1 invariant. Thus considering the totality of 

 transformations which leave 1 invariant, we see that the combina- 

 tion of any two transformations of the system are together equiva- 

 lent to a third transformation of the same system. Hence the 

 totality of transformations leaving a line invariant have the group 

 property and form a sub-group of the general projective group. 

 The same reasoning applies in general to the system of transfor- 

 mations leaving invariant any plane figure whatever. 



Theorem 5. All projective transformations of the plane leaving a 

 plane figure invariant have the group property and form a sub- 



