15 



/3" = ^cos I + a' sin I j /3' (^cos | - a' sin | j ; (j) 



and so on, for any number n of rotations. Let the last posi- 

 tion of j3 be denoted by j3n ; and since it can easily be proved, 

 by the theory of the multiplication of quaternions, that the 

 continued products which present themselves admit of being 

 thus transformed : 



^ cos __ + aC-') s>n -^)... (cos- + a sm - ) 



( a\ a„ a„ 



cos " + a sm — ) =: cos — + an sin — ; 

 2 2/ 2 ^ 2 



(o . a \ / a' , . a'\ 



cos a sm — I I cos — — a sm — I . . . 

 2 2 / \ 2 2 / 



/ a«-i , ., . a("-')\ a„ . a„ 



I cos a'-"~'>sm— - — l=cos— — a„ sm — ; 



(10 



in which a„ is a new vector unit, and a„ a new real angle, we 

 find that the result of all the n rotations is of the form 



^„ = (^cos -^ +an sm — j (i (^cos — — a„ sm — j. (1) 



It conducts, therefore, to the same final position which would 

 have been attained from the initial position j3, by a single 

 rotation = a„, round the single axis a„ ; the amount and axis 

 of this resultant rotation being determined by either of the 

 two equations of transformation (k), and being independent of 

 the direction of the line j3 which was operated on, so that they 

 are the same for all lines of the body. 



If the present results be combined with the theorem 

 marked (R), in the account, printed in the Proceedings of the 

 Academy, of the remarks made by the Author in Novem- 

 ber, 1843, it will at once be seen that if the several axes of 

 rotation be considered as terminating in the points of a sphe- 

 rical polygon, and if the angles of rotation be equal respec- 

 tively to the doubles of the angles of this polygon (and be 

 taken with proper signs or directions, determined by those 

 angles), then the total effects of all these rotations will vanish ; 



