XKWSON: PkOIKt'I'lVK IKANS^'OKM A IIONS. 75 



A ' E OP 



tlif ti^uif we see that ,^ i^— — -2-: setting these twci vahies o{ k' 

 AE AP^ 



AP OP OP 



equal to eacli otlier. we have . ^ . - ;^°°^ ^ .^," • Replacin"' 



A'P' b}- AP, and PP^ b}' AP^ - AP. this reduces to -^ - i, = 

 — — — . This gives a constant relation between P and P,, a pair of 



CO 



corresponding points. \\'e can now state the following theorem: 



Tlicorciii J. — A projective transforniatioii ^i'hi^h hai'cs //iiuina/i/ /70(> 

 ciu)icidcnt points at A transforms a /iv point P into P, s/(c/i t/iat tlw dif- 

 ference of the i-eeifrocals of tlie se^^nients tW and AP, is constant : tliis 

 constant is cQiial to tlie recifrocal of tlie sarnient AP , -toliei-e P is the 

 point -iCihicii eoi-responds to tlie point at infinity on 1. 



A transformation o{ this kind whicli leaves two coincident points 

 invariant will be designated b\' T),' . It is convenient to denote the 

 above constant 1)\' the same letter k which also denotes the conic 

 tletermining the transformation. Thus T{^ is a transformation de- 

 termined by a conic K: it leaves one and only one point A invariant, 

 the characteristic constant of the transformation, i. e., the reciprocal 

 of the segment AP^ is also k. 



Every conic touching the lines 1 and 1 ' determines a projective 

 transformation. It is therefore possible to construct as man^' dif- 

 ferent transformations of the points on the line 1 as there are conies 

 toucliing ] and 1.' A\'e know that oo'' conies can l)e drawn touching 

 any two lines: hence we infer that there are oo'^ projective trans- 

 formations of the points on a line. Among the oo-' conies touching 

 1 and 1 ' are oo- hyperbolas having one asvmptote perpendicular to 

 the line OX. Hence we infer that there are co- projective trans- 

 formations of the kind T' which leaves t\\o coincident points 

 invariant. 



Connected with every transformation of the kind T are three 

 quantities, the coordinates of the double points A and B (some 

 point on 1 being taken as an origin), and the characteristic anhar- 

 monic ratio k. These may be looked upon as independent variable 

 parameters, a consideration which again leads to the conclusion 

 that there are oo-' difterent projective transformations T of the 

 points on a line. Connected with ever\' transformation of the kind 

 1 ' there are two quantities, the coordinate of the invariant point 

 A, and the characteristic constant k. These ma\ be looked upon 

 as variable parameters of the transformation. * 



* In the analytic dev('l()i)meiit oi this topic Lios ('(uiation of t raiist'ormatinn S'lvt's 

 three iiidepeuucMii iJaraiiiciiMs, Lie clraily idcut ities t wd of liis parameters witli tlie 



