xewson: projfxtivf. transformations. 77 



. .T,ir=Tjj; where s=abc . .n. The characteristic anharmonic ratio 

 of the resultant transformation T^ is equal to the continued product 

 of the characteristic anharmonic ratios of the component trans- 

 formations T.^T]JTc. .Tq. The transformations of the s5'steni which 

 leave A and B invariant have therefore the property that the com- 

 bined effect of any two or more of them is equivalent to some 

 single one of the same system. In modern mathematical language 

 this continuous system cd' transformations forms a Continuous 

 Group of Transformations. 



This is a group of one parameter, the variable parameter of the 

 group being the characteristic anharmonic ratio. I shall denote 

 this group of transformations by the symbol G,. The subscript i 

 indicates that there are ooi distinct tradsformations in the group. 

 Sometimes the group will be designated by G ^^^ when it is desired 

 to call attention to the invariant points of the group. 



TlicorcDi J. — The totality of the projective traiisfoniiatioiis lu/iieh 

 ieavc two distinct points of a tine invariant forms a one-ternicd i:;roup 

 G,, whose fundamental property is that the combined effect produced l>v 

 tioo or more of the transformations of t/ie ,i:;roi/f is ecjuivalent to tliat oj 

 a single transformation of the same group. The characteristic anhar- 

 monic ratio of the resultant transformation is equal to tlie continued 

 product of the characteristic anharmonic ratios oJ tlie comfoue/it trans- 

 formations. 



Other properties of the group Gj will now be developed. 



If we have given any transformation T^^ of the group G^, we can 

 always find another of the same group that will exactly neutralize 

 the effect of the first. For if Tj, transforms P to P ' then k=( ABPP ' ). 

 If Tjj' be the transformation which transforms P' back to 



P, then k':^(ABP'P). But we know that (ABPP' )= , -^l, ^ ; 

 ' ■ ' (ABP'P) 



hence k= -^ , and k'= . Therefore k' exists whenever k ex- 

 k ' ]< 



ists. The transformations T|^ and Tj^/ are said to l)e inverse to one 

 another. The inverse of ever}- transformation belonging to the 

 group G, is also a transformation of the. same group. 



The relation to one another of the conies K and K,' \\hicli de- 

 termine a pair of inverse transformations, is easil}- seen to be very 

 simple. One of them is the reflection of the other on the line OX. 

 For if we project any range on 1 into a second range on ! ' by means 

 of a conic K, and then interchange 1 and I.' the conic K' which will 

 project the second range back into the first is obtained by revolving 

 k about OX through i8o°. The conies K and K' are both inscril)ed 

 in the same quadrilateral ABPj'.\.' 



