316 Mr Hart, On two models of parallel motions. [Dec. 8, 



Let ABCD be a quadrilateral (Fig. 1) such that AD = BG and 

 AB = CD. Let four points P, Q, 0, P'he taken on the four sides 

 AD, CD, BA, BC respectively, dividing them in the same ratio, i.e. 

 AP : AD=CQ '. CD = &c. 



Then it is evident that P, 0, Q, P are collinear, and that the 

 three lines AC, BD, P'OQP are parallel. It is further easily 



proved that 



AC.BD = AD'-AB\ 



or smce 



-4 7? A Ti 



AC = ^,.OP' and BD = ^.^.OP 

 BO AO 



OP. OP' = ^^^^^ {AD'-AB') 



= AD.AP-AO.BO. 



Suppose now the above four lines to represent four links jointed 

 at A, B, C, D, but otherwise free, then the four-point points P', 0, Q, P 

 will always be collinear, and their distances satisfy the condition 



P'0.0P[=P'Q.QP = P'0.P'Q = P'Q. QP] = AD.AP-A0.B0 



= a constant 

 = fi^ say. 



Thus if one of the points as be fixed and P' describe any 

 curve, then P describes its inverse. Constraining P' by a fifth link 

 OP' to describe a circle, P' also describes a circle, the magnitude 

 of which OP' being invariable depends on the distance between 

 the fixed points 0, 0'; if this = O'P', as in the figure, the circle de- 

 scribed by P' passes through 0, and the radius of its inverse becomes 



