490 Notices respecting New Books. 



parallel to that axis, and that any system of forces acting on a rigid 

 body can be reduced to a single force and a couple whose plane is 

 at right angles to the direction of the force. These theorems are 

 the starting-point of Theory. Adverting, in the first instance, to 

 the former, or kinematical theorem, it will be observed that the dis- 

 placement of the body might be effected by supposing it to be rigidly 

 attached to a screw working in a fixed nut, the axis of the screw 

 being properly directed and its pitch properly chosen. For the 

 purpose of the present discussion the term pitch denotes the distance 

 through which the screw advances when turned through the unit 

 angle of circular measure ; indeed a screw is defined by our author 

 to be " a straight line in space with which a definite linear magni- 

 tude, termed the pitch, is associated." Accordingly the theorem 

 may be stated thus : — any displacement of a rigid body may be re- 

 presented by a twist about a screw. In like manner, the second or 

 mechanical theorem above cited may be stated thus : — Any system 

 of forces acting on a rigid body may be reduced to a ivrench on a 

 screw, — meaning thereby a force directed along the axis of the screw 

 and a couple acting in a plane at right angles to the screw, whose 

 moment is the product of the force and the pitch. It is hardly 

 necessary to observe that the reductions contemplated in the.se two 

 theorems are unique. 



The composition of twists and wrenches can be effected by the 

 same method ; but we shall state it with reference to the former. 

 JLet the axes of x and y be two screws (a and ft) whose pitches are 

 p and q, and suppose the body to receive simultaneously about them 

 twists whose amplitudes are 6 cos I and 6 sin I, it can be shown 

 that these are equivalent to a single twist of amplitude about a 

 screw whose position is defined by the equations 



y—x tan I, (1) 



z = (p — q) sin I cos I, (2) 



and whose pitch is 



p cos 2 l + q sin 2 I (3) 



If we eliminate I between (1) and (2) we obtain 



zix> + if)-(p-q)xy = (4) 



This is, of course, the equation to the surface which is the locus of 

 the screws, the twists about which are the resultants of two twists, 

 of all relative magnitudes, simultaneously effected round a and ft. 

 It will be observed that each generator of the surface has its own 

 particular pitch given by (3), an expression w T hich can be represented 

 by an obvious geometrical construction. The surface is called the 

 cylindroid : a method of constructing a model of it is given on 

 p. 193 ; and a representation of the model when constructed forms 

 a frontispiece to the volume. 



The above method gives a rule for the composition in only a 

 particular case. It is completed, however, by observing that if we 

 have any two assigned screws (6 and <f) placed anyhow in space, 

 one cylindroid (and only one) can be drawn to contain them, and 



