( 212 ) 



by il variable logarithmic spiral. If the right line of the roiistant 

 direction is parallel to the ,r-axis, then the oscnlating s})iral becomes 

 circle of cnrvatnre (however ]iot remaining of constant size dnring 

 the motion). So the oscnlating descrii)tion of a cnrve (/) l\y a variable 

 circle of which the centre generates its evolntc, becomes a special case 

 of the oscnlating description by a constant logaritiimic spiral, of 

 which the pole W (determined according to 2) generates a definite 

 cnrve to be called an oh'iqve erohite of (/') (bccanse it is formed 

 bj the intersection of the snccessive right lines forming with the 

 snccessive tangents of (/') a constant obliqne angle). By changing the 

 oblique angle w^e obtain for one and the same cnrve (ƒ) an infinite 

 number of these obliqne evolutes. In contrast to this there. is only 

 one single trajectory of the pole of the variable logarithmic spiral in four- 

 point contact ; the pole V of this spiral is found in every position of 

 the system by projecting the describing point Qoï{f) on the tangent 

 of the rectifying curve in P. We wish to determine the taugentand 

 the radius of curvature (7, 8) of the oblique evolutes or trajectories 

 of the poles of the logarithmic spirals in tiireepoint contact and of 

 that of the spiral in fourpoint contact. Some investigations must 

 however precede concerning the motion of the line connecting Q 

 and V (6). 



(). To determine the point of contact ') of the right line Q V 

 (fig. 4) we notice that the motion of this i-ight liiie as invariable 

 system is determined by the motion of the point Q fbllo\ving the 

 describing point of (ƒ) and havino- thus a displacement equal to d.v 

 along SQ, and the condition QV 1. SP, must remain tangent to (7^). 

 For the latter it is necessary tliat the rotation of QV is equal to 

 that of SP; so we have first to determine the motion of SP (inva- 

 riable system determined by the motion of F as the describing point 

 of the evolute of (ƒ) (Ic)) and the contact of {F)). The motion of 



dv 

 SP results from t^vo rotations : the svstem rotation (h = — round 



y 



Q and the rotation c/s' of the radius of curvature MF round the 

 centre of curvature AT of the rectifying cnrve (F), which gives the 

 tangent SP its following position. So the momentary centre of the 

 resulting motion of SP lies on JlfQ; moreover P ha\ing in conse- 

 quence of this resulting motion to vovev the element of arc of the 

 evolute of (ƒ), that is having to undergo a displacement (h/ 1, AQ, 

 the momentary centre must also lie on Ff i.FQ and is thus the 



1) Point of inlerscclion of tlie right line QV with its following position. 



