112 KANSAS UNn'KRSll'V (jUARI'KkIA . 



Writing with Soplius Lie* a general infinitesimal projection of the 

 straight line b\- 



d,x=:=(aH bx-: cx-)d,t, (5) 



where x and t are independent variables, and a, b, c, constants, 

 we can substitute in equation (5) an expression containing tlie 

 values of the double points, ni and n, thus we get 



k (x — m) (x — n)=a-J-bx + cx-. 

 The differential equation (5) may therefore be written in the form: 



dx, j^ ,r 



rdt, (6) 



(x,— m)(x,~n' 



X, designating any variable in the transformed s}'stem and satisfy- 

 ing the condition, that for t^=o, Xj=x. The reason for this con- 

 dition is evident if it is remembered that for t~;o the identical 

 transformation results. 



The solution of tlu' differential equation (6) gives 



(me"»— ne™Mx4nin(eQit— e"t) , ., 



X,- -^ LSL^ *!: ^ (7) 



(e"^— emt)x-pmemt_ne"^ 



This e(}uation n-presents tlie onc-frr/iifJ grou}> of those transforma- 

 tions which lea\'e the points m and n invariant. Among tlie 

 transformations ot this group is also the involution leaving tlie 

 points m and n invariant, and they are determined b\' the condition 



uifi't ne"'^' -me'i" I ne"', or (m-n)e'"i-; -(m- n)e"', 

 or eCn- ii)t ^ 1 



Taking the logaritlims of both sides, we have 



(m — n)t log. ( — i). or finally t-~- '■ — ^- — , (8) where k ma\- be 



m — n 



anv real and integral number. Ecpiation (8) shows that in this 

 case there is no differential for t. i. e. , equation (6) has no meaning. 

 Hence there is only one involution among the projective trans- 

 formations leaving two points invariant. This result, well known 

 in geometrv. in connection witli equation ( 5) shows that there is no 

 infinitesimal transformation which b\- integration leads exclusiveh' 

 to involutions. 



Thus we ma\' sa\': 'jyic svs/t'/// of iin'olu/ioiis 011 a line has no 

 /iifiii/tfs/iiial fiutiisfonitatioii and, Iirncc. docs not form a group. 



Having thus shown that there are no involutoric transformations 

 transforming involutions into involutions we may ask whether there 

 are anv general projective transformations with this property, and 

 if so, find these transformations. 



*Vurlesjun,<j:oii. uobei' cont iiiuii'rlii'lie (!r'ui)i)eii. 



