( 213 ) 



poiiil of intersection U of .1/^^ widi /V. Tlic rolation of ^P round 



1/ Q'^f 



(his puinl is ^"^ — times llic rol;ili(»ii loiind Q, tli;it is tt.-. X ''^• 



This Siinie roUxdon must be ixMtbrmed hy the in\;iri;il»le sys(cm 

 (> F round its unknown momentary centre A', \\liilsl (^ is displaced 

 along AQ covering a distance ^ </.r = >/ <h. Fi-om the latter ensues 

 that the unknown momentary centre A' must lie on QP, where 



X(J\^ —'^y' (Is = 1/ (fe. From this we find 



XQ __ UM _ PM 



Therefore the point of contact R of Q Fis found by di-awing XR\_Q F 

 and as also PV i. QV, the above mentioned equation becomes 



RQ _ PM 

 VQ~DM' 

 So this is the equation which determines the i)Osition of the point 

 of contact R on Q F 



7. Trajectory (V) of the pole of the hnjnritltmic sp'urd in four- 

 point contact. ' 



a). Temgent. Let us describe a circle {N) through P, V and Q 

 (tig. 4), we can then regard this circle as a similar varying system 

 of which point P has a motion dij 1 SQ and point Q a motion dc 

 along SQ. The centre of the velocities of this motion is F, because 

 ^ VPP = Z ^^QQ' ^^"^^ ^^^'' V(2 = d!r.dr. This centre of the velo- 

 cities being situated on circle {X) itself, it is at the same lime one 

 of the points of contact of circle {N). Point I^ has in general dis- 

 placed itself along the circle in its second })osition ; the tangents of 

 the two positions in V ditfer inlinitesimally, so the tangent to the 

 trajectory ( F) is the tangent FT' to the circle {N] in F 



b). Raclius of curvature. To llnd the centre of curvature of 

 the trajectory ( I^) let us search for the })oint of intersection of 

 two consecutive normals iV F of tiiis trajectory. For that {)urpose 

 we shall consider Z\ FAY^. The vertex Q is displaced in the 

 direction QQ' , the vertex F according to the tangent FT', the vertex 

 JV, as a point of the similar system (A), in the directioji N'X', if 

 /^\'^XX'=Z \'^Q(^' . We can easily convince ourselves that these 

 thiee directions concur in one j)oint. So the triangle moves perspec- 

 tively ; so the points of contact of the sides lie in one I'ight line. 

 The [)oint of contact of X'Q (normal of the curve (ƒ)) is P (centre 

 of curvature of (ƒ)); the point of contact of QV is R (G). We then 



