288 THE THEORY OF SCREWS. [273, 



this expression by their sum; and, consequently, if p a be changed into 

 p a + x, and pp be changed into p$ x, the virtual coefficient will remain 

 unaltered whatever x may be. 



We have found, however ( 37), that the virtual coefficient admits of 

 representation in the form 



Pii/3i+ ... + profit- 



To augment the pitch of a by x, we substitute for a l} a..,, ... the several 

 values ( 265), 



SY&amp;gt; fit* 



a, + - cos a,!, a 2 + -^ cos a 2 , ... 

 2^] 2p 2 



where a 1( a 2 , ... are the angles made by the screw a with the screws of 

 reference. Similarly, to diminish the pitch of ft by x, we substitute for 

 &, /3 2 , . the several values 



OC OG 



&-H-COS&!, &-a- cos& 2 , &c. 



Zpi Zp 2 



With this change the virtual coefficient, as above expressed, becomes 



x \ ( Q x 



+ 9~ COS a - COS 



\ ^Pi 



or, 



CO 



o v/ ^j COS ttj + p 2 COS a 2 + ...! COS Oj 2 COS 2 . . . ) 



Z 



s ! cos &! cos a 2 cos & 2 cos a e cos 6, 



We have already shown that such a change must be void of effect upon 

 the virtual coefficient for all values of x. It therefore follows that the 

 coefficients of both x and x* in the expressions just written must be zero. 

 Hence we obtain the two following properties : 



= (p\ cos j + ... + /3 6 cos a 6 ) (^ cos h+ ... + a 6 cos 6 6 ), 

 _ cos a : cos bi cos a e cos 6 6 



The second of the two formulas is the important one for our present 

 purpose. It will be noted that though the two screws, a and @, are com 

 pletely arbitrary, yet the six direction cosines of a. with regard to the screws 

 of reference, and the six direction cosines of /3 with regard to the same 

 screws of reference, must be connected by this relation. Of course the 

 equation in this form is only true when the six screws of reference are 

 co-reciprocal. In the more general case the equivalent identity would be 

 of a much more complicated type. 



