64 VECTOR ANALYSIS. 



into which the dyadic may be expanded. We shall call it the 

 determinant of the dyadic, and shall denote it by the notation 



when the dyadic is expressed by a single letter. 



If a dyadic is incomplete, its determinant is zero, and conversely. 



The determinant of the product of any number of dyadics is 

 equal to the product of their determinants. The determinant of the 

 reciprocal of a dyadic is the reciprocal of the determinant of that 

 dyadic. The determinants of a dyadic and its conjugate are equal. 



The relation of the surfaces cr' and cr may be expressed by the equation 



141. Let us now consider the different cases of rotation and strain 

 as determined by the nature of the dyadic <. 

 If < is reducible to the form 



i,j> k, i',j'y h' being normal systems of unit vectors (see No. 11), the 

 body will suffer no change of form. For if 



we shall have 



p=xi'+yj'-\-zk'. 



Conversely, if the body suffers no change of form, the operating 

 dyadic is reducible to the above form. In such cases, it appears from 

 simple geometrical considerations that the displacement of the body 

 may be produced by a rotation about a certain axis. A dyadic 



reducible to the form 



i'i+j'j+k'k 

 may therefore be called a versor. 



142. The conjugate operator evidently produces the reverse rotation. 

 A versor, therefore, is the reciprocal of its conjugate. 



Conversely, if a dyadic is the reciprocal of its conjugate, it is either 

 a versor, or a versor multiplied by 1. For the dyadic may be 

 expressed in the form 



Its conjugate will be ia+j/3+ky. 



If these are reciprocals, we have 



But this relation cannot subsist unless a, ft, y are reciprocals to 

 themselves, i.e., unless they are mutually perpendicular unit-vectors. 

 Therefore, they either are a normal system of unit-vectors, or will 

 become such if their directions are reversed. Therefore, one of the 



dyadics 



ai + /3j + yk and ai /3j yk 

 is a versor. 



[See note on p. 90.] 



