NEWSON : ONE-DIMENSIONAL TRANSFORMATIONS. 125 



formation of the group. That this is its own inverse is self-evident. 

 The value k= — 1 gives the involutoric transformation of the group. 

 This transformation has the effect of interchanging every pair of cor- 

 responding points on the line, since its second power is the identical 

 transformation ; thus this transformation gives rise to an involution, 

 whence its name. 



The group Gi(AA') contains two very noteworthy transformations 

 whose cross-ratios are o and go, respectively. The first transforms all 

 points of the line except A into A'; the second transforms all points 

 of the line except A' into A. These are pseudo-transformations and 

 form an inverse pair. 



The cross-ratio of the identical transformation is unity, and this 

 transformation leaves every point of the line invariant. The trans- 

 formation of the group whose cross-ratio is 1 + d, when d is an in- 

 finitesimal number, moves every point on the line an infinitesimal 

 distance, and is hence called an infinitesimal transformation, d has 

 an infinite number of different values, viz., |8|exp.«'0. If an infinitesi- 

 mal transformation be repeated n times, the cross-ratio of the re- 

 sultant is (l + d) n . By a proper choice of d, i. e., of and n, this 

 cross-ratio may be made any number we please ; hence every trans- 

 formation in Gi(AA') may be generated from an infinitesimal trans- 

 formation of the group. The chief properties of the group Gi(AA') 

 may be summed up as follows: 



Theorem 9. The transformations of the group Gi(AA') can be ar- 

 ranged in inverse pairs ; it contains the identical transformation, one 

 involutoric, two pseudo and an infinite number of infinitesimal 

 transformations ; every transformation of the group may be generated 

 from an infinitesimal transformation of the group. 



One-parameter group Gi(A). — Let T and Ti be two transforma- 

 tions of type II having the same invariant point A. They may be 

 written : 



I7^T = T^A-+t and -A^=^ T + tl . 



T transforms x to xi, and Ti transforms xi to x-2. Their resultant, T2, 

 is obtained by eliminating xi from these two equations by addition, 

 giving us : 



irhr^T^r + t+k 



Thus, t2 = t + t. The resultant, T2, is of type II, has the same in- 

 variant point A, and its constant, t>, is equal to the sum of the con- 

 stants of T and Ti. 



The constant, t, may have an infinite number of values, and hence 

 there are an infinite number of transformations of type II having the 



