( 209 ) 



polar curve iisclf, wliieli Avill caiiso ils properlics lo he easily studied 

 and caleulatious of surface and leuglh of aic (o be executed in an 

 easier way. And it will he possible to trace in what a\ ay the consi- 

 dered tixed polar cur\'e can he described by other curves, the recti- 

 fying ones of which are likewise given (N". 3,4). Moreover the 

 investigation of these rectifying curves in a certain case (N". 12) 

 leads back to two kijids of spirals, fouiid already by Puiskux in 

 consequence of their tautochronisni for forces proportional to the 

 distance (Journal de Liouville, T. IX), but of svhich by this theory 

 more could be found about their geometrical pi'operties. 



Summing u[» in the following the chief })oints of my investigations 

 xQvy concisely I intend, if |>ossible, to revert to them more in detail. 



§ 1. jS/^otion of the recUfyuig curve; dinplest case. 



'I. Given in a movable plane an invariable sj'Stem (JS") consisting 

 of a riglit line AB (the ad'is fig. la) and a curve {F). The system 

 moves with the axis AB as movable polar curve. Let point Q of 

 this axis be the momentary pole, Q' the following, QP and Q' B' 

 1 AB. Let the elementary rotation da round Q' be taken of such 

 a dimension that the right line Q' B' coincides after the rotation 

 with Q' B regarded as a right line of the immovable plane ; let then 

 the rotation around Q" be taken in such a way that Q" B" coincides 

 Avith Q" B' etc. Then point Q describes a curve ( ƒ )(fig. '1 A) the locus 

 of the poles in the immovable plane, so the fixed polar curve 

 or the envelope of the axis AB in the immoA'able plane. 



We call {F) the rectifymn curve of (/) ; then (ƒ) itself is the 

 rectljied curve with respect to {F). 



The lines QB and Q' B' being two successive normals of the 

 curve (ƒ) cutting each other in P, the point B is the centre of 

 curvature of (/). 



If we assume in the system (2") the axis ^47i as ,r-axis and a right 

 line OY perpendicular to it as //-axis, we then immediatel^y see on 

 account of the nature of the generation of the curve: 



a. that the abscissae x of (J^) are the lengths of arc and the 

 ordinates y aro the radii of curvature of (/'), so that the rectifying 

 curve is at the same time the curve representing the radius of 

 ciirvatiu-e f> as a function of the arc s; 



b. that the elementary rotation of the system {2£) or the angle 



of contingency ol(/) is ds = — ; 



c. that the trajectory of point P moving along {F) in the immo- 



