2 TWISTS AND WRENCHES. 



certain standard position. We can have a movement 

 prescribed by which the body can be brought from the 

 standard position to the sought position. We have seen, 

 in the Introduction, that there is one preeminently simple 

 movement which will always answer. A certain axis can 

 be found, such that if the body be rotated around this axis 

 through a certain angle, and translated parallel to the 

 axis for a certain distance, the desired movement will be 

 effected. 



It will simplify the conception of the movement to 

 suppose, that at each epoch of the interval of time occu- 

 pied by the operations for producing the change of posi- 

 tion, the angle of rotation bears to the final angle of 

 rotation, the same ratio which the corresponding trans- 

 lation bears to the final translation. Under these cir- 

 cumstances the motion of the body is precisely the same 

 as if it were rigidly attached to the nut of a screw (in the 

 ordinary sense of the word), which had an appropriate 

 position in space, and an appropriate number of threads 

 to the inch. 



In order to express, in a scientific manner, the rela- 

 tion between the rotation and the translation in the 

 movement of a nut upon a screw, we give to the word 

 pitch a special meaning. We define the pitch to be the 

 rectilinear distance which the nut moves along the screw 

 when the nut is rotated through the angular unit of cir- 

 cular measure. The pitch is thus a linear magnitude. 

 The advantage of this convention is, that thejrectilinear 

 distance through which the nut moves when rotated 

 through a given angle is simply the product of the pitch 

 of the screw, and the circular measure of the angle. 



It will presently appear that screws have a dynamical 

 significance, which is of parallel importance to their 

 kinematical properties. For this reason we attach a 

 somewhat abstract sense to the word, by defining a screw 



