114 KANSAS UNIVF,RSri'V QUARTERI.Y. 



These X ^ projective transformations form a group, for 



bx— bk 



transformed bv 



^^ ckx-f-b 



bx, — bk, 

 x.,-- 



ck.x.+b 



, bk , bk 



bx-"b J 



b — ckk 

 x.,= 



b c k 1'; I 

 To each transformation can also be found the inverse in the same 

 s}'stem 



— bx.— bk . , f , 



^ — ' , or puttuig k lor k. 



ckx — b 



bx,— bk 



ckx , -i b 



The invariant points of this group are 



x = dri — , i- e.. elliptic if b and c are of the same sign. 

 c 



Taking the limiting case k x in e(]uation ( i i i the transformation 



becomes 



Transforming the general equation of involution b}- (14) we have: 



hex — ab 

 — cax — be 



Hence: TJicrc is one and oiilx one iiriU'liitoric traiisf(>]->iiation -(oliiili 

 /(•arcs the character of an iirvohttio)i itnchai/i^ci/. 



We will now give an example in which the connection between 

 the analytical method in this article and the s\'nthetical treatment 

 of the similar subject b\' Professor Newson in this number of 

 the QuAKiiCRi.v can easily be shown. Consider the special involu- 

 tion 



b 

 ■^' ex 



with the double points and 



c c 



After a transformation ( 11) this involution becomes: 



ckx — b 



x,= , . 



—ex — ek 



