CHAPTER IV. 



SCREW CO-ORDINATES. 



30. Introduction. We are accustomed, in ordinary sta- 

 tics, to resolve the forces acting on a rigid body into 

 three forces acting along given directions at a point 

 and three couples in three given planes. In the present 

 theory we are, however, led to regard a force as a wrench 

 on a screw, of which the pitch is zero, and a couple as a 

 wrench on a screw of which the pitch is infinite. The 

 familiar process just referred to is, therefore, only a 

 special case of the more general method of resolution by 

 which the intensities of the six wrenches on six given 

 screws can be determined, so that, when these wrenches 

 are compounded together, they shall constitute a wrench 

 of given intensity on a given screw.* 



The problem which has to be solved may be stated in 

 a more symmetrical manner as follows : 



To determine the intensities of the seven wrenches on 

 seven given screws, such that, when these wrenches are 

 applied to a rigid body, which is entirely free to move in 

 any way, they shall equilibrate. 



The solution of this problem is identical (15) with 

 that which may be enunciated as follows : 



To determine the amplitudes of seven small twists 

 about seven given screws, such that, if these twists be 



* If all the pitches be zero, the problem stated above reduces to the deter- 

 mination of the six forces along six given lines which shall be equivalent to a 

 given force. If further, the six lines of reference form the edges of a tetrahe- 

 dron, we have a problem which has been solved by Mobius, Crelle's Journal, 

 t. xviii., p. 207. 



