316] THE GEOMETRICAL THEORY. 341 



Let 77 be any screw on which an impulsive wrench is imparted, and let a 

 be the corresponding instantaneous screw, about which the body would have 

 begun to twist had it been free. 



Draw the cylindroid through a and X, and choose on this cylindroid the 

 screw /JL, which is reciprocal to p. 



Then ^ is the instantaneous screw about which the body commences to 

 twist, in consequence of the impulsive wrench on 77. 



For (77, p) is a cylindroid of impulsive screws, and (a, X) are the corre 

 sponding instantaneous screws. As p is reciprocal to p, it belongs to the 

 system of the fifth order. The corresponding impulsive screw must lie on 

 (77, p). The actual instantaneous motion could therefore have been produced 

 by impulsive wrenches on 77 and p. The latter would, however, be neutralized 

 by the reactions of the constraints. We therefore find that 77 is the impulsive 

 screw, corresponding to a as the instantaneous screw. 



316. Principal Screws of Inertia of Constrained Body. 



There is no more important theorem in this part of the Theory of Screws 

 than that which affirms that for a rigid body, with n degrees of freedom, 

 there are n screws, such that if the body when quiescent receives an 

 impulsive wrench about one of such screws, it will immediately commence 

 to move by twisting about the same screw. 



We shall show how the principles, already explained, will enable us to 

 construct these screws. 



We commence with the case in which the body has two degrees of 

 freedom. We take three screws, 77, , , arbitrarily selected on the 

 cylindroid, which expresses the freedom of the body. We can then de 

 termine, by the preceding investigation, the three instantaneous screws, 

 , ft, 7, on the same cylindroid, which correspond, respectively, to the 

 impulsive screws. Of course, if 77 happened to coincide with a, or with ft, 

 or f with 7, one of the principal screws of inertia would have been found. 

 But, in general, such pairs will not coincide. We have to show how, from 

 the knowledge of three such pairs, in general, the two principal screws of 

 inertia can be found. 



We employ the circular representation of the points on the cylindroid, 

 as explained in 50. The impulsive screws are represented by one system 

 of points, the corresponding instantaneous screws are represented by another 

 system of points. It is an essential principle, that the two systems of points, 

 so related, are homographic. The discovery of the principal screws of inertia 

 is thus reduced to the well-known problem of the discovery of the double 

 points of two homographic systems on a circle. 



