268 PROFESSOR TAIT ON THE ROTATION OF A 



V. eqaq- 1 = 2qY(Vq-iq.a)q- 1 



— 9.(V < r~ X q- a ~~ aYq— x q)q~ l 



— 9.(y*r~ 1 ?)2 ,— 1 - 9. a( T' 1 ~ qaqr 1 • q.(yi~ 1 q)<i~ l 



= Yqq~ l . qaq~ l — quq -1 . Yqq~ x 

 = 2V (Yqq- 1 .qaq- l ) 



the same equation as in § 7. 



9*. [Inserted Bee. 19th, 1868.] A geometrical investigation may also easily be 

 given, if for no other purpose than to serve as an instance of the justice of my 

 introductory remarks on diagrams as compared with quaternion equations. 



S^_0' 



Let Q, Q' be the poles, on the unit-sphere, of the versor angles BQE', BQ'E', 

 whose bounding arcs intersect in E' ; and let P, P' be the poles of these bounding 

 arcs, A the pole of QQ'B [A coincides with the projection of 0, the centre of the 



sphere]. Then evidently AP (=<?) and AP' ( = </) are the versor arcs, correspond- 

 ing to the above versor angles. Obviously the point E' is deduced from a point e 

 on the other side of the sphere [whose projection coincides with that of E'J, by a 

 rotation about Q through double of BQE', or about Q' through double of BQ'E'. 

 Hence we have obviously 



OE' = q Oeq~ l = q' Oeq'- 1 . 



