319] THE GEOMETRICAL THEORY. 347 



rigid body such that, if that body, while at rest and unconstrained, receive an 

 impulsive wrench about any screw of the first system, the instantaneous move 

 ment will be a twist about a screw of the second system. 



The two systems of corresponding impulsive and instantaneous screws 

 on the two systems of the third order, form two homographic systems. 

 There are, of course, infinite varieties in the possible homographic cor 

 respondences between the screws of two systems of the third order. The 

 number of such correspondences is just so many as the number of possible 

 homographic correspondences of points in two planes. There is, however, 

 only one correspondence which will fulfil the peculiar requirements when 

 one of the systems expresses the instantaneous screws, and the other the 

 impulsive screws severally corresponding thereto. 



If we are given one pair of corresponding impulsive and instantaneous 

 screws, the body is not by such data fully determined. We are only given 

 five coordinates, and four more remain, therefore, unknown. If we are 

 given two corresponding impulsive cylindroids and instantaneous cylindroids, 

 the body is still not completely specified. We have seen how eight of its 

 coordinates are determined, but there is still one remaining indeterminate. 

 If we are given a system of the fourth order of impulsive screws, and the 

 corresponding system of the fourth order of instantaneous screws, the body, 

 as in the other cases, remaining perfectly free, there are also, as we shall see 

 in the next section, a singly infinite number of rigid bodies which fulfil the 

 necessary conditions. In like manner, it will appear that, if we are given a 

 system of the fifth order consisting of impulsive screws, and a corresponding 

 system of the fifth order consisting of instantaneous screws, the body has 

 really as much indeterminateness as if we had only been given a single 

 pair of corresponding screws. 



But the case of two systems of the third order is exceptional, in that 

 when it is known that one of these is the locus of the instantaneous screws, 

 which correspond to the screws of the other system regarded as impulsive 

 screws, the rigid body for which this state of things is possible is completely 

 and uniquely specified as to each and every one of its nine coordinates. 



319. A Property of Reciprocal Screw Systems. 



Given a system of the fourth order A and another system of the fourth 

 order P. If it be known that the latter is the locus of the screws on which 

 must lie the impulsive wrenches which would, if applied to a free rigid 

 body, cause instantaneous twist velocities about the several screws on A, 

 let us consider what can be inferred as to the rigid body from this fact alone. 



Let A be the cylmdroid which is composed of the screws reciprocal to A. 



