276 THE THEORY OF SCREWS. [261- 



The functions thus arising are well known as &quot; emanants &quot; in the theory 

 of modern algebra. The cases which we shall consider are those of n = 1 and 

 n = 2. In the former case the emanant may be written 



- df n df 



262. Angle between Two Screws. 



It will of course be understood that/ is perfectly arbitrary, but results of 

 interest may be most reasonably anticipated when / has been chosen with 

 special relevancy to the Dyname itself, as distinguished from the influence 

 due merely to the screws of reference. We shall first take for / the square 

 of the intensity of the Dyname, the expression for which is found ( 35) 

 to be 



where (12) denotes the cosine of the angle between the first and second 

 screws of reference, which are here taken to be perfectly arbitrary. The 

 second group of reference screws we shall take in a special form. They are 

 to be a canonical co-reciprocal system, so that 



R = (\ + X 2 ) 2 + (\ 3 + X 4 ) 2 + (X 5 + X 6 ) 2 . 

 Introducing these values, we have, as the first emanant, 



X 2 ) + (p a 4- /i 4 ) (X 3 + X 4 ) + (fjL 6 + fi t ) (X 5 + X 6 ) ; 



but in the latter form the expression obviously denotes the cosine of the 

 angle between a and /3 where the intensities are both unity ; hence, whatever 

 be the screws of reference, we must have for the cosine of the angle between 

 the two screws the result 



263. Screws at Right Angles. 



In general we have the following formula for the cosine of the angle 

 between two Dynames multiplied into the product of their intensities : 



dR n dR dR 



This expression, equated to zero, gives the condition that the two Dynames 

 be rectangular. 



