92 KANSAS UNIVERSITY QUARTERLY. 



We must examine the two transformations corresponding to X-^o 

 and X-=oo . We learned in one-dimensional groups to call these 

 psuedo-transformations. In order to understand the psuedo-trans- 

 formations in the plane we must consider the value of the constant 

 a. First let a be positive between o and i; second let a be positive 

 between i and x : third let a be negative. Let ABC be the invari- 

 ant triangle; and let the cliaracteristic anharmonic ratio along BC 

 be A, along CA be X"-'-; and along AB be X'^"* all Uken in the same 

 order around the triangle. For X -^o and a a positive fraction these 

 ratios are respectively o, oo , oo. Hence (Kansas University Q'Jar- 

 terly Vol. lY, page 79) all points of the plane except the line AB 

 are transformed into the point C: the line AB is indeterminately 

 transformed. For X--x and a a positive fraction the values of these 

 ratios are respectively ex, o, o. Hence all points of the plane ex- 

 cept the points on the line AC are transformed to B; the points on 

 AC are indeterminately transformed. In the second place let a be 

 positive between i, and oc; for X= o the three anharmonic ratios in 

 the order mentioned above are o, cc, o. Hence in this case all 

 points of the plane except the line AB are transformed to the point 

 C. For X=- 00 and a between i and cc tlie vahus of the -three ratios 

 are respectively ex, o, 00. Hence all points of the plane are trans- 

 formed to A except those on BC. In the third place let a be nega- 

 tive; for X~ o the three values are respectively o, o, 00. Hence all 

 points except those on BC are transformed to A. For X^^ 00 the 

 values are respectively cc, oc, o; hence all points of the plane are 

 transformed to B except those on AC. In general it can be shown 

 that when a is any complex quantity for X o and for X cc all 

 points of the plane are transformed into some one of the invariant 

 points. 



Prop. 4. The group G to?i fains t7i'o pS(iui'o-(raf;sfcrn:atu?is 7i /,<sr 

 charaitrristic anharmonic ratios arc o and cc rcspciiivcly. A psrudo- 

 transformatioti transforms all points of tJie plane to one of the invariant 

 points except the opposite- side of the invariant triangle ; this is indeter 

 minately transformed. 



The group G contains an identical transformation for which X — i. 

 Let X^=( i+S) where 8 is infinitesimally small. Since the identical 

 transformation transforms each point of the plane into itself, it is 

 evident that the infinitesimal transformation moves ever}' point of the 

 plane an infinitesimal amount. If this infinitesimal transformation 

 be repeated n times it will give rise to a transformation of the group 

 X=(i-f8)". When n=oo and S is a complex infinitesimal, X can 

 be made to ecpial any finite real or complex, quantity by a proper 



