162 PHILLIPS AND MOORE. 



If the characteristic equation has a pair of roots equal to X + i yu 

 it will have a second pair equal to X — i /j,. There will then be a 

 pencil of imaginary lines corresponding to X H-i /x and another pencil 

 corresponding to X — i fx. The imaginary planes in which these pencils 

 lie are left invariant. Any plane through the origin cutting each of 

 these invariant planes in a line will contain two invariant lines and so 

 will be invariant. In particular the oo2 real planes cutting the planes 

 of the invariant pencils in lines will be invariant. Thus repeated 

 imaginary roots characterize the motions which have oo ~ real invari- 

 ant planes discussed by Moore. 



3. Canonical form of the dyadic. Let ki and ^2 be perpendic- 

 ular unit vectors. The dyadic 



"^ = (ki ki + k2 k2) cos q + (^'2 ^'1 — A'l k2) sin q. (5) 



represents a rotation of the plane ku though the angle q measured 

 from ki toward ko. That is if the vector r is rotated into r' 



r' = ^-r. 



This is the most general form of a proper rotation of the plane into 

 itself. Any improper rotation can be expressed in the form 



X — {ki ki — ko ^2) cos q -\- (k2 ki + /ci A-o) sin q. (6) 



By properly choosing ki and /v"2 this can always be reduced to a simpler 

 form. Let 



k( = ki cos I Q' + ^2 sin I q, 



ko = kx sin \ q — ko cos | q. 



By direct expansion it is seen that 



k[ k[ — ko ko = (ki A'l — ko ko) cos q + (k2 ki + ki ki) sin q. 



If then we use /;(, ko, instead of ku ko any improper rotation of the 

 plane into itself can be expressed in the form 



X = ki ki — ko ko. (7) 



Let 



kij,i= 1, 3,...2«-l,i= i + 1, (8) 



be a set of n completely perpendicular planes and let ki, kj be per- 

 pendicular unit vectors in the plane ki,-. The dyadic 



^ = S (ki ki =*= kj kj) cos qa -\- {kj ki =f ki kj) sin qij, (9) 



