CHAPTER I. 



TWISTS AND WRENCHES. 



1. Definition of the word Pitch. 



The direct problem offered by the Dynamics of a Rigid Body may be 

 thus stated. To determine at any instant the position of a rigid body 

 subjected to certain constraints and acted upon by certain forces. We may 

 first inquire as to the manner in which the solution of this problem ought to 

 be presented. Adopting one position of the body as a standard of reference, 

 a complete solution of the problem ought to provide the means of deriving 

 the position at any epoch from the standard position. We are thus led to 

 inquire into the most convenient method of specifying one position of a body 

 with respect to another. 



To make our course plain let us consider the case of a mathematical point. 

 To define the position of the point P with reference to a standard point A, 

 there can be no more simple method than to indicate the straight line along 

 which it would be necessary for a particle to travel from A in order to arrive 

 at P, as well as the length of the journey. There is a more general 

 method of defining the position of a rigid body with reference to a certain 

 standard position. We can have a movement prescribed by which the body 

 can be brought from the standard position to the sought position. It was 

 shown in the Introduction that there is one simple movement which will 

 always answer. A certain axis can be found, such that if the body be rotated 

 around this axis through a determinate angle, and translated parallel to the 

 axis for a determinate distance, the desired movement will be effected. 



It will simplify the conception of the movement to suppose, that at each 

 epoch of the time occupied in the operations producing the change of position, 

 the angle of rotation bears to the final angle of rotation, the same ratio 

 which the corresponding translation bears to the final translation. Under 

 these circumstances the motion of the body is precisely the same as if it were 

 attached to the nut of a uniform screw (in the ordinary sense of the word), 



