674 MOORE. 



which, if we put <:i2 = 1, gives for the invariant planes 



(38) P = A-12 + Cn(kn ± A'42) + Cu{h4 + A-23) ± (C132 + Cu^)hi 



Therefore, // the rates of rotation in the jjlanes Mi aiid M2 are different 

 the only %tlanes left invariant by (33) are Mi and Mo but if the rates of 

 rotation in the tivo planes are the same or differ only in sign then a two 

 parameter family of planes u'hich belongs to a three parameter linear 

 system of eomplexes is left invariant}'^ 



The planes of the system (38) all cut the planes 



(A'i3± k^) + i{ku^ A-23), (A*i3=*= A-42) — i{hi^ hz). 



These are the planes mentioned above. 



We saw that in case mi= ^m2 the complex M can be resolved in 

 00 2 different ways into the sum of two completely perpendicular planes. 

 The pairs of planes belong to the set (38). The transformation (34) 

 can then be represented in 00 2 different ways as the sum of rotations 

 parallel to pairs of completely perpendicular planes. The rates of 

 rotation parallel to both planes of a pair are the same but different 

 for different pairs. 



The above set of invariant planes were found under the condition 

 that their magnitude be left unchanged. We might however have 



12 In the article referred to in note 2 Cole states the theorem "Every rota- 

 tion in a four dimensional space for which ?? 5^ 0. (The condition here would be^ 

 that neither wi nor jno is zero) can be reduced to a succession of two simple 

 rotations whose fixed planes are absolutely perpendicular to each other. 

 This decomposition can be effected in only one way." From the above 

 theorem it is evident that this statement is inaccurate. He discussed finite 

 rotations and writes the equations of the rotation 



x' = X cos 6 — y sin d, y' = xsind + y cos 6 

 z' = z cos ^ — oj sin <p, w' = 2'sin <p + w'cos <p. 



He states that the invariant planes of this rotation are the xy- and gio-planes. 

 This is true oi d 9-^ tp. But ii 6 = <p any line passing through and lying in 

 the i)lancs x + iy = 0, z + ra- = is left invariant also any line passing 

 through and lying in the plane .r — iy = 0, 2"— iw = is left invariant. 

 Hence any plane containing two of these invariant lines will be kept invariant. 

 There is a two parameter family of these planes which are real and therefore 

 00- real planes are left invariant. If = —(p then every line passing through 

 and lying in one of the jjlanes x + iy = 0, 2 — ico = or .r — iy = 0, 

 2 + ico = is left invariant and all the planes passing through and cutting 

 each of those iilanes in a line is left invariant. In the first case it is easy to 

 see that the plane x = z, y = w is left invariant and in the second case x = w, 

 y = z is left invariant. The error in Cole's work arose from the fact that in 

 determining the coordinates of the invariant planes he failed to take into 

 account that it was possible for all the denominators of his expressions to 

 vanish simultancoasly. 



