On the Rotation of a Rigid Body about a Fixed Point. 441 



taken at the node in which the two planes intersect, and then 

 the points A, B swing into their new positions A', B' by 

 means of a rotation about the pole of the great circle, of which 

 A ; B' forms a part. This mode of effecting the displacement 

 naturally suggests the consideration of the effect of rotations 

 taking place successively about two axes fixed in space. 



First, then, let us study the effect of the combination of a 

 rotation (a) having P for its pole, followed by another (ft), of 

 which Q is the pole, P and Q being points in the surface of 

 the revolving sphere. 



In drawing the annexed fi- 

 gure, I have supposed that the 

 two rotations are of the same 

 kind, each tending, when a 

 spectator is standing with his 

 head to the respective poles and 

 his feet to the centre, to make 

 a point to his right-hand pass 

 i?i front qf his face towards his 

 left-hand. Let now PQ revolve 



through - positively into the 



position of PR, and through ^ negatively into that of QR. 



Then I say that the two impressed rotations a and j3 about P 

 and Q will be equivalent to a single rotation about R, equal 

 to twice the acute angle between QR, RP. 



Let the first rotation about P bring Q to Q' and R to R' ; 

 it is clear that QPR, Q'PR, Q'PR' are all equal triangles. 

 Therefore R'Q'R = 2PQR < ==/3. Consequently the positive 

 rotation ft about Q' (the new position of R) will carry R ; back 

 again to R, its original position. Hence the actual motion 

 which results from the successive rotations combined being 

 consistent with R remaining at rest, must be equivalent to a 

 single rotation about R. 



To find its magnitude, let the second rotation carry P to 

 P'* ; then the angular displacement PRP' (which is the re- 

 quired rotation of the whole body) is equal to twice the acute 

 angle between Q'R, RP, which is the same as that between 

 QR, RP, as was to be shown. Thus we see that the semi- 

 rotations about three poles (considered as the angular points 

 of a spherical triangle), which, taken in order, would bring 

 the sphere back to its first undisturbed position, are equal to 

 the included angles at such poles respectively. 



* The reader is requested to fill in the point P' and join P'R. 



