[ 187 ] 



XXV. On the Geometrical Theory of Rotation. By W. F. 

 DoNKiN, M.A. ^'c, Savilian Professor of Astronomy in the 

 University of Oxford"^. 



ALTHOUGH two demonstrations t have already appeared in 

 this Journal of the triangle of rotations (to adopt Mr. Syl- 

 vester's convenient designation), I think it worth while to add 

 the following, which, though not substantially different, exhibits 

 the theorem in the simplest way, and under the most striking 

 aspect. I shall also subjoin some fiu-ther illustrations of its use 

 in connexion with quaternions. Fig. 1 . 



Let ABC, fig. 1, be 

 any triangle on a sphere 

 fixed in space, and a/37 

 a triangle on an equal 

 and concentric sphere, 

 moveable about its 

 centre. The sides and 

 angles of a/37 ^^^ equal 

 to those of ABC, but 

 differently arranged,one 

 triangle being the in- 

 verse or reflexion of the 

 other. [In the figm-e 

 straight lines are used 

 for convenience to re- 

 present arcs of great 

 circles.] , 



Now if the triangle a)87 be placed m the position 1, so that 

 the sides containing the angle a maybe in the same great circl^ 

 mth those containing A, it is obvious that it may slide along AB 

 into the position 2, and then along BC into the position 3 ; into 

 which last position it might also be brought by shding along AC. 

 Hence, denoting the rotations by arcs on the fixed sphere, we 

 have the following theorem : — 



L Twice the rotation AB followed by twice the rotation BC 

 produces the same displacement us twice tlie rotation AC. 

 Or, denoting them by arcs on the moving sphere, — 

 Twice the rotation /3« followed by twice tlui rotation 7/3 produces 

 the same disjjlacement as twice the rotation 7a. , u r* 



Again, in fig. 2, if we denote the centre of the sphere by O, 

 it is obvious that the pyramid whose intersection with the sur- 

 face of the sphere is the triangle u^% might pass from position 1 

 into position 2, by turning round OA through an angle 27r— 3A, 



* Commniiicated hv the Author. 



t See Phil. Mag. for June aud December 1850, and January 1851 . 



