( 2J7 ) 



J 3. Panihola on the (td'i.s (ts rccti/ii'nui curve. For the [)ai';iliola 

 (lig. 9) the centre is at iiifniily ; llic coiisidemtioiis about llie point 

 R based on (liis centre do nol liold good liere. If we delennine the 

 radius of curvature of the cvolute as a special case of a pohir trajectory 

 of a Ihreepoiut logarithmic spiral (8) by drawing (f^Kl tangent /M^ 

 it is exident according to the i)roperties of the [)arahola thai this 

 radius of curvature remains constant, ecpial to p, parameter of llie 

 l)araI)ola, because PI represents the length of the subnormal. So the 

 evolute is a circle and the rectitied curve (ƒ) an evolvent of the 

 circle. Point R (point of contact of Q V) coincides here with /, because 

 / is a tixed i)oint of 1^ So it is situated here too on the right line 

 (/V) comiecting / with the centre of the })aral)ola. 



,J4. T<iatoclironisni. The condition that a motion along a giveji 

 curxe be tautochronous is: the tajigential component of the force must 

 be proportional to the length of the air between the moving point 

 to a point of the curve; in that case the motion takes place as a 

 single oscillatory motion. For the curves whose rectifving curves are 

 central conies (JO, 12) where the force is supposed to act from the 

 tixed point /?, IIL (fig. 7, 8) is proportional to QO as li' to cM.JO). 

 In order that the motion along those curves I)e tautochronous with i) as 

 centre, the tangential component of the force must be pro|)orlional 

 to RL, so the force itself (directed along /i^) proportional to /t (2, that 

 is to the distance. So for a force acting from the centre R in pro- 

 [)ortioii to the distance both curves are tautochronous. liut the centre 

 of tautochroiiism is to l)e reached along the curve only in the 

 cases of circle, ellipse or hyperbola (,i'-axis imaginary) as rectifying 

 curves, so only cycloid, epi- and hypocycloid and the spiral of the 

 second kind (rectifying curve a hyperbola on the imaginar}' axis) 

 are in reality tautochrone ; for the spiral of the tirst kind the centre 

 of tautochronism does not lie on the curve, for the evolvent of the 

 circle it lies at infinite distance. For the epicycloid the force must 

 repel; for the hypocycloid and the s})iial of the second kind it must 

 attract. 



For the cycloid point 11 lies at intlnile distaiice ; the force becomes 

 constant and is directed according to the tangent in a cns[>. 



