498 



PROCEEDINGS OF THE AMERICAN ACADEMY. 



If $ is any dyadic, then we have seen that a'$ is another 1-vector. 

 In general a** is not equal to ^^a. If, however, $ is self conjugate, 

 a.$ = $'a; and if $ is anti-self conjugate a'$ = — <l>'a. Hence it 

 may readily be shown that an anti-selfconjugate dyadic turns a vector 

 into a perpendicular vector. 



The d^'adic which turns a vector into itself is called the idemfactor I. 

 Thus 



a. I = I. a = a; (154) 



for I is selfconjugate, and when expanded in terms of chosen coordi- 

 nate vectors is,^^ in the non-Euclidean geometry which we are dis- 

 cussing, 



I = kiki + koko + ksks — k4k4. 



62. We could now proceed to develop the theory of dyadics in- 

 volving vectors of any dimensionalities and their products with each 

 other and with vectors of various dimensionalities. In general 

 if a, /3, 7 are vectors of an^' dimensionalities the dyad /?7 may be defined 

 in terms of our inner product by the equation a* 037) = (a •,3)7. This 

 product is itself a d^^ad unless a, /? are of the same dimensionality. 

 Such a discussion, however, would carry us further than is necessary 

 for our present purpose, and we shall therefore consider chiefly one 

 case, which has acquired particular importance through the work of 

 Minkowski. 



If r is any 1-vector, and A any 2-vector, then the product 



r' = r«A 



is a linear ^'ector function of r. It is evident therefore that this 

 multiplication by A is equivalent to a multiplication by some dyadic 

 fi. Let us find the relation between this dyadic Q, and A. 



If $ is any dyadic (made up of 1-vectors), we may define the prod- 

 ucts <J>'A and A'<J> by first defining the products, 



(ab)-A= a(b-A) 



A«(ab) = (A- a) b, 



67 As a matrix the idemfactor would be written 



and the laws of multiplication of matrices would be modified. It is possible, 

 however, to keep the ordinary theory of matrices by the introduction of 

 imaginaries, as Minkowski does. 



