gz KANSAS l;n"i\krsi r\- (^)Iak ^^:Kl,^•. 



peiidicular to OX: It-t a tangent be drawn cutting 1 and 1 ' in 1:' and 

 P.' It is easy to sliow that the difference' of the segments OP and 

 OP' is constant tor all tangents to the jiarabola: and that this 

 constant difference is equal to the distance from O to the point of 

 contact with 1. The effect of a transformation of this kind is to 

 move every point of the line to the right or to the left a certain 

 fixed distance. It is equivalent to sliding the line along itself: 

 thereby the length of an}- segment is unaltered. A transformation 

 of this kind is called a Translaii(>ii. It is self evident that all trans- 

 lations form a group. 



llu'orriit 1 6. -llic fraiislaliojis i>f thr fioiii/s t>/i a ///ir toi''ii <J <'»<'- 

 tn-Difd 'j^foiip oj thf fv/^r (t [ : ///,■ x/;/i^-/i- iiirariant p,>int (>f f/i/s i:;r(>iip is 

 thr f^oiiii (if i/ifiii/fv i>/i the liiir. 



We shall next examine the one-termed sub-group whose invariant 

 points are O and cc. This one-termed group is common to the 

 two- termed perspective grouj) (i._>,) and to the two-termed dilative 

 group G-'co: we ^hall therefore expect to find in it a combination of 

 the projH'rtit'S ol these two groups. An\' transformation of this 

 group is constructed by drawing a pt luil of lines from a point Q on 

 the line at inffnitw The lines ot such a pencil are of course paral- 

 lel. The transformations effected h\ the s\-stems of parallel lines 

 form a group, which is both perspective and dilative. 



Thus far wi' ha\c considered the groups of transf(U"mations of the 

 line into itselt. Other one dimensional forms remain to be con- 

 sidered e. g. a pencil of lines through a i)oint O and a pencil of 

 planes through a line 1. The tluors' for these forms is so nearly 

 identical with that for the straight line that it is •unnecessar\' to 

 give the parallel development. 



A perfecth' ol)vious construction lor a juojective transformation 

 of a pencil of ra\s is as follows: Let two planes PI and P'l meet 

 in a line 1: and let O, a point on 1, be the common vertex of two 

 pencils of ra\s. oiu- in PI and tlu' other in P'l. The planes de- 

 termined by three pairs of corresponding lines of the two pencils 

 together with PI and P'l determine a cone of the second order 

 having its vertex at O and touching both PI and P'l. Tangent 

 planes to this cone cut PI and P'l in corresponding lines of the 

 two pencils. If now P ' 1 be revolved about 1 until it coincides with 

 PI, a projective transformation of the pencil in PI is completed. 

 The properties of this transformation and of groups of such trans- 

 formations ma\' easih' be developed. 



