492 Notices respecting New Books. 



to determine the reciprocal screw complex, which involves the theory 

 of equilibrium. The next group of questions will be those which 

 relate to the effect of an impulse upon a quiescent rigid body, free 

 to twist about all the screws of the screw complex. Finally, we 

 shall discuss the small oscillations of a rigid body in the vicinity of 

 a position of stable equilibrium, under the influence of a given 

 system of forces, the movements of the body being limited as before 

 to the screws of the screw complex" (p. 83). 



An explanation of the meaning of the technical terms printed in 

 italics is needed for the elucidation of this passage. In the first 

 place suppose a body to be free to receive a twist about each of n 

 screws, and let it receive any twist about each of them, its final 

 position could have been reached by a twist round some screw which 

 is merely one of an infinite number about which it is possible for 

 the body to twist. All these screws together with the original n 

 screws is called a screw complex of the nth order; and if a body can- 

 not be twisted about any screw but one belonging to this complex, 

 it is said to have freedom of the nth order. It is to be observed 

 that n cannot exceed six ; so that a body can only have six different 

 orders of freedom. Now if we suppose a screw to be reciprocal to 

 n of the screws of a screw complex of the nth order, it will be re- 

 ciprocal to every screw of the complex. Aud all these screws will 

 form the reciprocal screw complex. If the screw complex is of the 

 nth order, the reciprocal screw complex is of the (6— n)th order. 

 Of course, if the freedom of the body is expressed by any assigned 

 screw complex, that body will remain at rest when acted on by a 

 wrench round any screw of the reciprocal screw complex. 



Consider, for instance, the case of the equilibrium of a body which 

 has freedom of the second order. Jjn this case the body is free to 

 twist round two screws and <£>, and therefore round any screw of 

 the cylindroid (0, 0). This cylindroid is therefore the screw complex 

 of the second order. Through each point of space (as we have seen) 

 a cone of reciprocal screws can be drawn. The author speaks of 

 each of these cones as a reciprocal screw complex of the fourth 

 order, one for each point in space ; but, of course, all of them to- 

 gether form the reciprocal screw complex of the fourth order ac- 

 cording to the definition above cited. If any one of these cones is 

 considered, each generating line has its own proper pitch, and the 

 pitches have every value from + oo to — go ; so that one screw of 

 given pitch belonging to a screw complex of the fourth order can be 

 drawn through each point in space. Of course it follows, from what 

 has been stated above, that if a body, being at rest and having 

 freedom of the second order, receive a wrench round any one of 

 these screws, it will remain at rest. Suppose a force P to act on 

 the body along an assigned line passing through a given point, and 

 let the pitch proper to this line be p ; the body will continue at 

 rest under the action of P and a couple which has a moment 

 P p and acts in a plane at right angles to the line. Thus, using 

 the same axes of coordinates as in equation (4), the body will be 

 at rest under the action of any couple whose plane is parallel to 



