WILSON: REDUCTION OF REAL DYADICS 177 



The proof here given differs radically, however, from that given 

 by Gibbs 5 for the simple three dimensional case; it applies, 

 moreover, to any pair of conjugate latent roots, simple or not, 

 when shearing terms are absent. 



In case there is a double complex root with shearing, the 

 terms in the reduced complex form of the dyadic are 



aa | a + ay \ y' + a \ y' + bj3 \ $' + 65 | ti + | ti 



Multiplication shows that the corresponding real form is, in 

 matrical notation, as follows: 



s cos 9 s sin 1 



— s sin 6 s cos 6 1 



s cos 6 s sin 6 



— s sin 6 s cos d 



The extension to the case of multiple roots with various 

 shearing terms is clear. In the matrix there are two-rowed 

 determinants strung along the main diagonal all alike; and 

 parallel to the main diagonal there are strung along with any 

 distribution (depending on the distribution of the original 

 shearing terms) two -rowed determinants, all alike, and of the 

 special form shown above; all other places are filled with zeros. 

 The transformation in the case of multiply complex roots 

 with shearing might be called a cyclotonic shear. It consists 

 of a stretch and of an elliptic rotation in a series of planes Pi, 

 P 2 , . . . , Pa-ij Pk} the angles of rotation and the factors of 

 stretching being the same for all, combined with a shift of the 

 points in P A+1 parallel to P h for at least some values of h. The 

 amount of the shift is typical of the shear. For instance, in 

 the case of a triple complex root with double shearing the 

 vector 



p = Xa y + ya 2 + 27i + ^72 + ^«l + ^«2 



suffers, in addition to the stretch and elliptical rotation, the shift 



+ Zcti + Wa 2 + M7i + #72 • 

 ' Gibbs-Wilson. Vector Analysis, p. 360. 



