I 4 6 KINEMATICS. [268. 



age with equal, but intersecting, opposite sides, which we may 

 call anti-parallelogram (Fig. 64). If 3 4 be fixed, the instanta- 

 neous centre of i 2 is the intersection 5 of 4 i and 2 3. 



Fig. 64. 



To obtain the centrodes in this case, notice that as the tri- 

 angles 152 and 534 are equal, the triangle 5 4 2 is isosceles ; 

 hence 51 = 53, and 45 35=41=0. The difference of the 

 radii vectores of 5 drawn from 4 and 3 being thus constant, it 

 follows that the space centrode is a hyperbola whose foci are 

 4, 3, and whose real axis =a. As 43 = 12 = ^, the equation of 

 this hyperbola is 



for 4 3 as axis of x and the midpoint of 4 3 as origin. 



It is easy to see that the space centrode becomes an ellipse 

 when b < a. 



As the triangles 152 and 354 are equal the body centrode is 

 an equal hyperbola or ellipse. The two centrodes lie symmet- 

 rically with respect to their common tangent at 5. 



For a given anti-parallelogram the centrodes are hyperbolas 

 when one of the larger links is fixed ; they are ellipses when 

 one of the shorter links is fixed. 



268. If in the anti-parallelogram only one point, say 4, be 

 fixed, it can be used as an inversor, i.e. as an instrument for 

 describing the inverse of a given curve 



