176 WILSON: REDUCTION OF REAL DYADICS 



(d') If a and b are conjugate, so are $ and $&, and Z a and Zb. 

 If Zo is reduced to a certain standard form, one form of the con- 

 jugate imaginary dyadic Zb will be that in which each vector 

 (antecedent or consequent) and each scalar in Z a is replaced by 

 its conjugate value. Hence the types of Z a and Zb or of <£> a and 

 <£& must be identical relative to the distribution of shearing terms. 



The results thus obtained allow us to set up canonical forms 

 for real dyadics which have imaginary latent roots. As the ante- 

 cedents a, /3, ... of the dyads occur in conjugate imaginary 

 pairs, the consequents which form the reciprocal set a , /3', . . . 

 (p. 8) also occur in conjugate imaginary pairs (since they are ob- 

 tained by multiplication and division). 



If there is a pair of simple roots, the corresponding terms in 

 the reduced form of the dyadic are a a \ a + b /3 | $' where 



a ' a = 0' = 1, a /3 = p' a = 0, 



owing to the relations between reciprocal sets. We may write 



a = a { + a 2 i b = a x — a 2 i 



a = «! + a 2 i j3 = a { — a 2 i 



I „• al I • 



with the reciprocal relations yielding 



o i o i o on 



«l aj -f- a 2 a 2 = 1 «i a 2 — a 2 a x = U 



&i «j — a 2 oc 2 = U «i «2 "T «2 a i = " 



when real and imaginary parts are separated. Hence 

 o o n ooi 



If we set a[= 2a° u a 2 = 2a° 2 , the sets a, /3, . . . and a', /3', . . . 

 may be replaced bya b a 2 , . . . and a{, a 2 , . . . 

 On multiplying, the terms a a \ a + b p | j3' give 



a i a i I <*i + %*i | a 2 s COS0 a x | a[ + S sin0 a x \ a! 2 



i or 



a 2 a 2 I a i+ «i«2 I «2 — S Sin0 a 2 \ a\ + S COS0 a 2 | a 2 



if a = se*. This is precisely of the Gibbs cyclotonic form, as 

 might have been anticipated. The linear transformation or 

 strain is a combination of stretching with elliptical rotation. 4 



4 An elliptical rotation of angle q is a projection of an ordinary rotation of 

 angle q. See Vector Analysis, p. 349. 



