268.] 



PLANE MOTION. 



Let r OP be the radius vector drawn from an arbitrary fixed 

 origin, or pole, O to a given curve ; on OP lay off a length 



= r' = K 2 /r, where tc is a constant; then P 1 is said to 

 describe the inverse of the given curve. 



The theory of inversors is based on the following geometrical 

 proposition: if three lines CA=a, CA'=a, CO = b (Fig. 65) 

 turn about C so that O, A, A' 

 are always in line, the product 

 OA OA' remains constant, viz. 

 OA OA ' = b 2 a 2 . For if the 

 circle of radius a described about 

 C intersect the line OC in B and 

 B 1 we have OA-OA* = OB-OB' 



This proposition shows that in the anti-parallelogram 1234 



(Fig. 66), with the vertex 4 fixed, the line joining the vertices 4 



and 2 intersects the circle described about 3 with radius 3 2 in a 



I point 2 r such that 2 and 2' describe inverse curves with respect 



to 4 as pole. For we have 4 2^-4 2 = 4 3 2 2 3 2 =& 2 a 2 . 



Fig. 66. 



Moreover, any parallel to 42 will intersect the links 41, 43, 

 2 i in points O, A, A' dividing the three lines in the same ratio; 

 hence if O be fixed, A and A' will describe inverse curves for 

 O as pole. This is the principle on which Hart's inversor is 

 based. 



