CHAPTER XVII. 



FREEDOM OF THE FIFTH ORDER. 



229. Screw Reciprocal to Five Screws. 



There is no more important theorem in the Theory of Screws than that 

 which asserts the existence of one screw reciprocal to five given screws. 

 At the commencement, therefore, of the chapter of which this theorem is 

 the foundation, it may be well to give a demonstration founded on elementary 

 principles. 



Let one of the five given screws be typified by 



j tch _ 



fc Pk 7* 



while the desired screw is defined by 



(pitch = p). 



x x _y y __ z - z 



P 7 



The condition of reciprocity ( 20) produces five equations of the following 

 type : 



k)7k 

 From these five equations the relative values of the six quantities 



can be determined by linear solution. Introducing these values into the 

 identity 



a. (yy (3z } + (az yx) 4 y (fix ay ) = 0, 



gives the equation which determines p. 



