( 214 ) 



find the point of T^^V, in other words tlie required centre of curva- 

 ture Mv of {V) as the point of intersection of 'NV witii PR. 



8. Trajectory {W) of tlie pole of the constant hiiarltlunlc sinral 

 in tltreepoint contact. 



a). Tangent. Let the rectifying curve of tlie logarithmic spiral l>e 

 Sa^P (fig. 5), pole W. 



Angle PScvQ remaining constant the triangle PWQ forms during 

 the whole motion a similar varying system. Of this system T^, the 

 pole of the spiral in fourpoint contact (7), is the cejitre of the velocities, 

 l)ecause /VPP' = /_VQQ' and VP: VQ = (h/:dc. So the vertex 

 W of A PWQ moves in such a way that /VW]V' = /^VPP'. 

 As P, W, V, Q are concyclic it is easy to see that If' 11^' lies iji the 

 production of QW. So QW is the tangent to the trajectory (IT). 



b). Radius of curvature. To tind the centre of cur\utui"e of the 

 trajectory (IF) we must tind the point of contact of the normal 

 PW of this trajectory, that is that point of PW of which the 

 motion is directed according to PW itself, if we regard this right 

 line again as a right line of the similar varying system QWP. 

 This point is found by letting down VM^ out of the centre of velo- 

 cities T^ in such a way that /_VM,,S,:= /^VPP' , ov^V.]ftoP= 

 supplement of /_VPP' =: /^VIP. So /, M^,., V, P lie on a circle 

 and ^/FP being a right angle /^IM^^.P is a right one too. So the 

 desired centre of cui-vature J/„. is found by producing QV ± SrP 

 till it intersects PI in / and by letting down a perpendicular IM ,, 

 out of I on to Sa- P. 



§ 3. Conies on their aves as rectfyiny curves. 



9. As a means for the treatment of the conies as rectifying curves 

 let us first regard the right line PN (fig. 6), where P in the system 

 motion of (iJ) describes the e volute of (ƒ) whilst N is a fixed point 

 of the ,v-axis and let us then determine its point of contact. The 

 right line PX of which the motion is determined by the motion 

 of P and N {P describes the evolute, j,Vone of the evolvents of (ƒ) (1)), 

 can be regarded as a similar system ; point P has a motion = r/y 1 AX; 

 point X, as a consec^uence of the system rotation = </? about Q, a 

 motion =: QX X '^^ likewise j_ AX. So the |)oint of contact T in 

 question lies on FX in such a way that 



TX QNXde' 



