CHAPTER XIII. 



THE DYNAMICS OF A RIGID BODY WHICH HAS FREEDOM 

 OF THE FIFTH ORDER. 



151. Screw Reciprocal to Five Screws. There is no 

 tnore 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 elemen- 

 tary principles. 



Let one of the five given screws be typified by 



x-*k y-yk z-Zk, . 



= -r = - - (pitch = pa), 



while the desired screw is defined by 



x 



-x\ y-V z- z' , 



-y _ (pitch 



The condition of reciprocity (22) produces five equa- 

 tions of the following type : 



o[(p + pk)a k + ykyk - )3*z*] + /3[(p + pkjfik + a k z k - 

 + yf(p + pk)yk + fikXk - a k yk] + a k (yy' - j32 x ) + (3 k (az' - 

 + yk (fix* - ay) - o. 



From these five equations the relative values of the 

 six quantities 



can be determined by linear solution. Introducing these 

 values into the identity 



M 



