( 216 ) 

 and for the nidiiis of the ti^ed one 



Moreover 



R = RT = — ; = — . 



R b Ji h 



— ==: en 



R -y2r a 



For (I ::= h (cirele) R becomes equal to x , the rectified cui\ e 

 equal to a cijchid. 



J 2. Hijpcrbola on one of its a,ves as rectify 'ukj curve. Let us 

 tirst take the real axis as the axis of the arcs (fig. 8). As in 10 it 

 is evident, that the point of intersection R of OP and QV '\^ Si 



fixed i)oint during the whole of the motion, \\ here — r=: 



^ ° RP h'' 



(2« and 2^y real and imaginarv axis of the hyperbola). The point T (10) 

 is imaginary here ; instead of this we consider the constant three- 

 point logarithmic spiral whose rectifying curve WP is parallel to 

 one of the asymptotes of the hyperbola. 



Let us project both R and W (pole of the logarithmic spiral) 

 on PQ, it then follows from the equation PR : RO =^ fr : d'-, that 



h^ 

 the quotient of the projections of PR and RO is equal to — , and 



from the rectangular triangle PMQ, where PW: H'f^ = fj:((, that 

 the quotient of the projections of P]r and 117^ is likewise equal 

 to /r:a^. So the projections of R and IT coincide in L. So the 

 tangent W(} of the trajectory (IT) (8) forms a constant angle 

 ]V'WR = / ]VQü with the radius vector /ilT; so the ti'ajectory 

 (IF) is .a logarithmic spiral of the same shape as the constant 

 describing logarithmic spiral (117^, that is the curve (ƒ) is described 

 by a constant logarithmic spiral which moves in threepoint contact 

 ^vitll itself in such a way that its pole describes the same logarithmic 

 spiral with opposite curvature. 



Allowing for the modification of the figure we find that these 

 considerations are literally the same for the hyperl)ola on the 

 imaginar}' axis as the axis of the arcs. Of the additional geometric 

 considerations to which the two kinds of spirals whose rectifying 

 curves are hyperbolae give rise, we shall mention only that the 

 two kinds of spirals are each other's evolntes and that both of them 

 approach asymptotically logarithmic spirals of a definite position, 

 with which they have a fourfold contact at infinity (the rectifying 

 curves being the asymptotes of the hyperbola). 



