newson: continuous groups. 85 



AZ AZ' BX BX' CY CV 



kz= : ; k^.= : ; and ky^ : . 



BZ BZ' CX CX' ■ AY AY' 



But by similar right triangles we have 



AZ Aa AZ' Aa' BX Bb BX' Bb' CY Cc CY' Cc' 



BZ Bb' BZ' Bb'' CX Cc CX' Cc'' AY Aa ' AY' Aa' 

 Multiplying together and substituting we get 



AZ BZ' BX CX' CY CY' 



BZ AZ' CX BX' AY AY' 



Aa Bb' Bb Cc' Cc Aa' 



Bb Aa' Cc Bb' Aa Cc' 



Theorem 6. Every projective traiisfonnation of the kiint T in 

 the plane determines a eliaraeteristic anharmonie ratio along 

 each of the invariant lines and through each of the invariant points. 

 When these three anharmonie ratios are reckoned in the same order 

 around the triangle their product is unity. 



Thus we see that of these three anharmonie ratios only two are 

 independent. Every transformation of T depends therefore upon 8 

 parameters, viz: the six co-ordinates of the three invariant points 

 (or lines) and these two independent anharmonie ratios. Since 

 each of these parameters may assume oa^ different values, we see 

 again that there are 00'^ transformations of the kind T in the plane. 



We are also enabled to distinguish two distinct varieties of varia- 

 ble parameters, viz: co-ordinates of invariant points (or lines) and 

 characteristic anharmonie ratios. This is an important distinction 

 which will be of considerable use later on. 



Theorem y. Of the eight parameters ichich deter/nine a transforma- 

 tion of the kind T six are coordinates of invariant points {or lines) and 

 two are characteristic anharmonie ratios. 



We proceed now to consider the system of transformations leav- 

 ing a triangle invariant. In this case the six co-ordinate parame- 

 ters are constant and the two anharmonie ratio parameters are vari- 

 able; thus we see that there are oo^ transformations leaving a given 

 triangle invariant. From another point of view we arrive at the 

 same result. The two conies K and K^ by means of which we can 

 construct the transformation T touch four fixed lines I, x, y, z. K 

 and Ri therefore belong to a range of co^ conies touching the same 

 four lines. Any pair of conies taken from this range determines a 

 transformation leaving the triangle (ABC) invariant. 00- pairs of 

 conies may be formed from this range, thus showing that there are 



