Notices respecting New Books. 491 



which they determine, the pitch corresponding to each generating 

 line (it must be remembered) being also determined. It can be 

 shown that if a body receive twists about three screws on a cylin- 

 droid, and if the amplitude of the twist about any one screw is pro- 

 portional to the sine of the angle between the other two screws, the 

 body will return to its original position. Hence, supposing the 

 cyhndroid determined by two screws to be constructed, we have a 

 simple general rule for the composition of two twists about those 

 screws. And the like is true of wrenches. 



Let there be two screws whose pitches are p and q inclined to 

 each other at an angle O, and let d denote the shortest distance 

 between them ; suppose the body to receive a wrench of intensity 

 P round the latter and to twist round the former through an angle 

 of small amplitude a ; the work can be proved to equal 



a P {(j^ + ^cos O — d sinOj- : 



the coefficient of a, P is called the virtual coefficient of the two 

 screws. It is plain that if the constants are so related that 



(p + q) cos O — d sin O=0 

 no work will be done by P, and consequently that the two screws 

 p and q have this property, that a body at rest, but free to twist 

 round the former, remains at rest when it receives a wrench round 

 the latter. The form of the above expression shows that if the body 

 is at rest, but free to twist round the latter screw (q), it will remain 

 at rest when it receives a wrench round the former (p). Two 

 screws (p and q) thus related are called reciprocal screws. 



Now suppose a screw (77) to be reciprocal to two screws (0 and <p\ 

 it will be reciprocal to every screw in the cylmdroid (0, <p) ; -q will 

 cut the cylindroid in three points, and therefore will meet three 

 screws, to one of which it will be perpendicular ; and the other two 

 will have pitches equal and of opposite sign to its own. Let any 

 point P be taken, and from it let lines be drawn at right angles to 

 the screws of a cylmdroid severally; any one line meets the surface 

 on two other screws of equal pitches ; let a pitch equal and of oppo- 

 site sign to them be attributed to it ; the locus of the lines (which 

 are proved to form a cone of the second order) is the reciprocal cone 

 drawn through that point to the cylindroid. 



The points we have hitherto mentioned relate to the foundation 

 of the method, and receive considerable development before it is 

 applied to the discussion of the dynamics of a body enjoying dif- 

 ferent orders of fredom. How the method is applied may be in 

 some degree understood from the following general statement re- 

 garding the contents of the latter part of the volume. In the dis- 

 cussion of each order of freedom " we shall first ascertain" (says the 

 author) M what can be learned as to the kinematics of a rigid body, 

 so far as small displacements are concerned, from merely knowing 

 the order of the freedom which is permitted by the constraints. This 

 will conduct us to a knowledge of the screw complete, which exactly 

 defines the freedom enjoyed by the body. We shall then be enabled 



