( 2J0 ) 



vable plane is tlie cvohite of (ƒ) of which the length of arc is 

 found on the ordinate of {F); 



d. that the Irajectoiy of an arbitrary fixed point C of AB 

 is one of the evolvents of (ƒ) starting from that point of (ƒ), 

 which becomes the pole in the immovable i)lane when (2 is in C ; 



e. that the area of the figure, comprised between the rectified 

 curve (ƒ), its evolute (P) and two of its radii of curvature, is lialf 

 of the area of the figure between (7^), the axis AB and the corre- 

 sponding ordinates. 



2. Right line as rectifying curve. Let the rectifying curve be the 

 right line AB (lig. 2) and let the motion have advanced as far as 

 the pole Q, centre of curvature of (ƒ) P. The following motion is 

 an elementary rotation dt- = ^ P' Q' P ov /^ QBQ' round Q' . If 

 we let fall out of Q and Q' jjerpendiculars QV and Q' V' on AB, 

 then at the limit the points V and Q' lie on the circle, described 

 on PQ as a diameter. So ^ QVQ' z= /^ QPQ', consequently also 

 Z VQ' V'=/_PQ'P', the elementary rotation. Farthermore ^ Q' V' A 

 being a right angle the system rotation round Q' causes point T' to 

 arrive in V represented as a point of the immovable plane. The 

 same holds good for the following rotations. So the (variable) pi'o- 

 jcction V of Q on AB in the immovable plane is a fixed point. 

 As moreover the angle VQA (angle of the tangent of the rectified 

 curve with the radius vector out of T") remains constant, the 

 rectified curve is a logarithmic spiral with V for pole. 



The trajectory of the pole of the logarithmic spiral in the movable 

 system is the right line AJ). So when a logarithmic spiral rolls over 

 one of its tangents, its [)ole describes a right line. 



The place of the pole in the movable system is found for ever}^ 

 moment by projecting the corresponding mouientar}' centre Q of 

 the motion on the right line {F). The part (2-1 of the .r-axis 

 corresponds to the arc of the logarithmic spiral, which approaching 

 the pole, winds round it in an infinite number of revolutions; the point 

 A of the ,i'-axis is unattainable by this arc ; QA is (he limit of the 

 length of arc of Q measured towards the pole. The sha|)e of the 

 logarithmic spiral depends exclusively upon one datum : the angle 

 of the right line (7^^) with the .ivaxis. 



As a special case there is the right line parallel to the ./'-axis as 

 rectifying curve : the rectified one becomes a circle (logarilhinic 

 spiral where the angle between radius vector and tangent is a right 

 one ; the polo of the s|)ii-al becomes the centre of the circle.) 



