180 PHILLIPS AND MOORE, 



and 



^23* ^31= bn 



respectively, both of which are correct. Fomiula (40) is then correct 

 for all unit two-vectors and since it is distributive it is therefore true 

 when TTi, TTo, ITS, ir^ are complex two vectors. 



11. Rotation of a vector. Let 



where tti and X2 are two completely perpendicular unit planes in 

 terms of which M is linearly expressible. Suppose <^ rotates a unit 

 vector r through an angle 6 into a unit vector 



Then 



cos d = r • r' = r-(f)-r. (42) 



Let a be the angle between r and the invariant plane n. We can 

 then write 



r = ri+ r2 



where ri and r2 are the projections of r on the invariant planes tti and 

 7r2. Since <^Ti and </)-r2 are in the planes tti, and 7r2 respectively 



and so 



Hence 



ri-0-r2= 0, 



COS0 = r-0-r = ri-0-ri+ r2-(^-r2 



= r^ cos^ a cos ^1+ r^ sin^ a cos 52- 



cos d = cos^Q! cos 91+ sin^a cos q2. (43) 



The angle 6 then depends only on a, qi and 92- Since tti is invariant r 

 rotates into a vector r' making the same angle a with tti. Conse- 

 quently the motion rotates / through the same angle 6 that it rotates r 

 through. 



If gi and ^2 are not equal it is clear from (43) that the maximum and 



minimum valves of 6 are determined by a = 0, - , that is, when r lies 



