274 MR G. UDNY YULE OX THE ASSOCIATION 



or replacing K by the original Q, 

 (AB) = i + Q(i+ , + **)^~- SSQv, - Q*U - V - V) (7). 



24. The question arises, what is the meaning of the alternative sign in the 

 expression for (AB) ? One of the values given is, as a matter of fact, only a numerical 

 solution, and is really impossible, so that the value of (AB) is not indeterminate. We 

 may write (7) in the form 



and f^ must be less than 1 and greater than 0. 

 (A) 



The product of the two values given by the + and sign is 



2Q (!+,) 



If Q be negative this is negative, and so the lower value is negative or impossible ; 

 we must consequently use the + sign. On the other hand, if we subtract 1 from 

 each value above and again form the product, it is 



and this is negative if Q be positive ; hence one of the values is greater than unity. 

 When Q is positive we must therefore use the sign to the radical. 



If Q = + 1, one of the roots is unity and the other greater or less, according as 

 S.J 5 s,. Thus, if Q = -f- 1 we have for (AB) 



or 



i.e., (AB) = (A) or (B). 



If Q is 1 on the other hand, one root is zero 



(AB) = or ( -f(s 1 +* 2 ). 



^ 25. The values of all four groups are as follows, the first sign of the radical to be 

 used when Q is positive : 



(AB) = 



4Q 



