lyo KANSAS UNIVERSITY (^)UARTERLV. 



6. An exactly similar tlieory holds for one-termed groups of 

 real projective transformations of the most general form in two, 

 three and n dimensions. The parameter of every such group is an 

 anharmonic ratio. Such a group contains one identical, one in- 

 volutoric, two pseudo and two infinitesimal transformations. The 

 transformations of the group whose characteristic anharmonic 

 ratios are negative can not be generated b}^ either infinitesimal 

 transformation. 



It would seem at first sight that these results contradict Lie's 

 general theorem that every transformation of the projective group, 

 in any number of variables, can be generated by the repetition of 

 an infinitesimal projective transformation. (See "Cont. Gruppen " 

 page 45.) Lie's theorem is correct for the group of all real and 

 imaginary transformations; but as has just been shown it is not true 

 that every real transformation can be generated b}' the repeti- 

 tion of a real infinitesimal transformation. 



I have learned recently from a note in the Bulletin of the New 

 York Mathematical Society for Nov. "93, page 66, -that Professors 

 Study of Marburg and Engel of Leipsic have reached a similar re- 

 sult for the case of Lie's special linear homogenous group in two 

 variables; but I do not know that they are yet in possession of the 

 complete theory. Professor Taber of Clark University has quite 

 recently established nearly the same results in the special cases of 

 the group of orthogonal transformations (New York Bulletin for 

 July '94, page 258-9) and the group whose invariant is an alternate 

 bilinear form (Mathematische Annalen, Band 46, page 581, Nov. 

 '95.) Professor Taber has not given his results their obvious 

 geometric meaning. 



A new proof of Lie's theorem is here outlined. Let (i-f dz), 

 where dz is an infinitely small complex quantity, be the character- 

 istic anharmonic ratio of an infinitesimal transformation. If it 

 can be shown that (i-(-dz)n=k, where \\:^a::> and k is any real or 

 complex number, then Lie's theorem is proved; for the assumption 

 of complex values of k and real or imaginary invariant space ele- 

 ments is equivalent to the assumption of complex values of the 



variables and constants in the equation J =^k '- • We may 



X, — n X — n 



set (i-|-dz)=re^y; then y is infinitely small and r is infinitesimally 

 less or greater than, or equal to, unity. This value can be repre- 

 sented in the complex plane by a point infinitely near to the unit 

 point. The locus of a point representing the value of r°e™>' where 

 n becomes infinite is a spiral, which by a proper choice of dz can 

 be made to pass through any point of the plane. 



