15 
PS plea (cos 5+ a’ sin 5) sf (cos 57 a’ sin 5) (j) 
and so on, for any number x of rotations. Let the last posi- 
tion of (3 be denoted by 3, ; 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 : 
a—) a1) a’ a’ 
cos a®@—) sin habe (cos ; sing ) 
( “4 = +a’ sin 
an 
2 3 
(cose +a sinS =o + an sin 
wi +-§ —— 
a -¥q a’ 1% xa (k) 
(oss — asin >) (cos F —a sins). 
at 4 al——) an - Gn 
( cos Waa MUR ow 3 ) =eo8% Say sin 3 
in which a, is a new vector unit, and a, a new real angle, we 
find that the result of all the m rotations is of the form 
—— (cos + ap sin 3) pB (cos — ap sin - (1) 
It conducts, therefore, to the same final position which would 
have been attained from the initial position B, 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 (3 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; 
* 
