160 QUATEENIONS IN THE ALGEBRA OF VECTORS. 



extended to matrices of any order) affords a point of departure from 

 which the properties of matrices may be deduced with the utmost 

 facility. The ordinary matricular product is expressed by a dot, as 

 $.ty. Other important kinds of multiplication may be defined by 

 the equations 



With these definitions <I>x <&<]? will be the determinant of 3?, and 

 <l>x3? will be the conjugate of the reciprocal of <3> multiplied by twice 

 the determinant. If < represents the manner in which vectors are 

 affected by a strain, ^3>$$ will represent the manner in which 

 surfaces are affected, and <l?x < 3>:<l > ^ ne manner in which volumes 

 are affected. Considerations of this kind do not attach themselves so 

 naturally to the notation = aSX+/3S// + ySi>, nor does the subject 

 admit so free a development with this notation, principally because 

 the symbol S refers to a special use of the matrix, and is very much 

 in the way when we want to apply the matrix to other uses, or to 

 subject it to various operations. 



