ROTATIONS IN SPACE OF EVEN DIMENSIONS. 165 



4. Exponential form of dyadic. If ^ is any dyadic, we shall 

 write 



\I>2 -Jrn 



6'* = / + ^ + --+ .... +^+ ... (13) 



where I is the identical dyadic. The sum of n terms in (13) is a 

 dyadic. That this sum approaches a definite limit, when 71 increases 

 indefinitely has been showTi by various writers. ^^ By direct expan- 

 sion it is seen that 



g^qii {ki kj - ki ki) = 2 {ki hi + hj kj) cos qu + (A-, kj - ki kj) sin qij. 



Hence the dyadic representing any proper rotation can be written 

 in the form 



(f, = 0^<lii (kjki-kiki) (^\/^ 



According to the inner multiplication of Lewis 



fC j ' rC ij /i if 



krkij = 0, I ^ i,j. 

 The identical dyadic is 



I = A'l A'l -p A,"2 ko ~\~ • ■ ■ -kin kin- 



Hence 



I-kij = kjki — kikj. (15) 



The product being obtained by multiplying the consequents of / with 

 kij. If then we write 



M = ^qiikii, (16) 



equation (14) takes the form 



<l>=e'-^. (17) 



This is a form in which the dyadic representing any proper rotation 

 in 2n — dimensional space can be written. 



In a space of 2/i + 1 dimensions the dyadic representing a given 

 rotation has a characteristic equation of the form 



(^2n + 1 -I- ai 02" + . . . . a2„ <^ + a2„ + 1 = 0. 

 15 Cf . H. B. Phillips. Functions of Matrices, Amer. J. of Math. Vol. XLI. 



