( •211 ) 



§ 2. }[ovahle and rariahh rectifying curves. 



3. If two curves (/') and (ƒ') osculate each other in a point Qix\\{\ 

 if the evolute (/>') of the latter is allowed to I'oll over the evolnte 

 {]>) of the former, the cnrve (/) which does not move, will be the 

 envelope of the moving cnr\e (ƒ') desci'ibed osculatingly by it in 

 all points; the point of contact Q displacing itself along the moving 

 curve (/'') describes the fixed curve (ƒ). 



Let us take of (ƒ) and {/') the rectifying curves [F) and {F') 

 (Fig. 3), then for the first condition (osculating each other in Q) the 

 .r-axes of these rectifying curves must coincide and the two rectifying 

 curves must intersect each other in P perpendicularly over Q. The 

 following ordinate (radius of curvature of (/') ) 1^' iQ' i of {F') is 

 equal to the ordinate (radius of curvature of ( ƒ ) ) l\Qi of {F). To 

 make these rays of curvature coincide a displacement of the sj'Stem 

 of the rectifying curves {F') is necessary over a distance Q\Q^ 



So the above-mentioned osculating description of a curve (/) by 

 another curve (ƒ') corresponds to the description of its rectifying 

 curve {F) by the rectifying curve {F') by means of a pai'allel displa- 

 cement of {F') parallel to the .I'-axis ; the variable point of intersection 

 F on {F') describes the curve {F). The amount of the elementary 

 displacement Q\Q^z=z(li' — dv,' is determined l\y the diirerence of 

 the abscis-elements (/,/' and (/,// uiiich correspond in both curves to 

 the increase of the coinciding ordinate y to the following ordinate 



y + ^^y- 



4. When the rectifying curve i F' ) does not intersect the rectifying 

 curve {F) but touches it, the rectitied curve (/') has the following 

 radius of curvature in common with (/) which it touches by a 

 contact of the third order (four consecutive points in common). If 

 we allow {F) to be described envelopingly by tiie rectifying cur\e 

 {F') which then not only changes its positioji but also its sha[)e 

 according to a definite law, then this corresponds to the description 

 in fourpoint contact of the rectified curve (ƒ) by the variable and 

 moving rectified curve {/')• 



The evolute of (/) is described osculatijigly by the Nariablc and 

 moving evolute of (ƒ'); the evolute of the evolute of (/) is enveloped 

 by that of (ƒ') in two[)oint contact. 



5. In particular an arbiti-ary rectifying curve can be described 

 intersectingly by a right line of constant direction or tangcntially 

 by a right line of variable direction ; this is (2) every curve in 

 threepoint contact by a constant logarithmic spiral or in fourpoint contact 



