Equation of Motion of Dynamics of Continuous Media. 795 

 A form of (/> which occurs frequently is : 



0. = A^. + BJ. + dk., .... (2) 



where i, j, and k are the three fundamental unit vectors. 



This form is not less general than that given in (1), for 

 all forms of $ may be reduced to that of (2). 



When Ai, B^ and C x are of the forms 



Bi = pyxi+pwi+pyzk ; 



C] = pz X i+pzyi+pzzk; 

 in which 



Pxy = Pyx, 'p xg = p zx , p yz = p zy , 



<p. is self-conjugate, and the operation 



< /). = iA 1 .+jB 1 . + kC 1 (2') 



When </>. is not self-conjugate there is a second function, <j> f , 

 called the conjugate of </>> which possesses the property 



A.</>.B = B.f.A (3) 



This point has been explained here, because it is on this 

 simple property that the transformations depend. A complete 

 account is given in any text-book on Vector Algebra *. 



If we are dealing with four dimensions we shall have, 

 corresponding to (2), the operator 



<l> = A 1 i 1 + A 2 i 2 + A 3 i3-f-A4i4, 



where the i's denote the four fundamental unit vectors. 

 These are mutually perpendicular, so that i 1 .i 2 : =0 i etc. 

 Corresponding to (3), we have in this case 



A.<I>.B = B.<I>'.A, (30 



where A and B are four-dimensional vectors. 



A function, <j>, occurs in the equations of the mechanics of 

 continuous media, and the jo's denote the stress components. 

 This particular function is found to be self-conjugate as a 

 result of the principle of moments, or we have the relations 



Pxy=Pw etc \ 



In four dimensions we have a simple example of a 



* E. g., ' Vectorial Mechanics/ ch. v., Silberstein; ' Vector Analysis,' 

 Wilson. 



2G2 



