188 Prof. Donkin on the Geometrical Theory of Rotation. 



and then into position 3 by turning Fig. 2. 



round OB througli an angle 27r— 2B; 



into which position it might also have 



been brought from 1 by turning 



roimd OC (in the contraiy direction) 



through an angle Stt— 2C. Hence 



we have what may be called, with 



reference to theorem I., the j^olar 



theorem; namely, as regards the fixed 



sphere, — 



II. A positive rotation 2(7r— A) 

 round OA, followed by a positive rota- 

 tion 2(7r— B) round OB, produces the same displacement as a nega- 

 tive rotation 2("7r — C) round OC. 



The enunciation of the theorem is the same as regards the 

 moving sphere, if a/37 ^^ P^^^ foi' ABC. But it must be carefully 

 observed, that on the fixed sphere the arrangement of ABC is 

 such that C is the positive pole of a rotation from CA to CB ; 

 whilst, on the moving sphere, 7 is the negative pole of a rotation 

 from 7« to y^. 



It is obvious that a perfectly similar demonstration may be 

 employed in the case of a polygon. As regards fig. 2, this has 

 been given by Mr. Sylvester. Also, if we take the moving 

 polygon equilateral but not equiangular with the fixed polygon, 

 and then diminish the sides indefinitely, we arrive, in the case of 

 fig. 2, at Poinsot's mode of repi'esenting the most general kind 

 of rotatoiy motion ; whilst in the case of fig. 1, taking the poly- 

 gons equiangular but not equilateral, we obtain two cones, the 

 reciprocals of Poinsot's cones, one of which slides {withont rolling) 

 upon the other. Thus the two theorems given above are, in fact, 

 particular cases of the general relations between the cui'vatm'es 

 of these cones and the angular velocities of the body about the 

 instantaneous axis, and of the instantaneous axis in the body or 

 in space. 



In the Philosophical Magazine for June 1850, I showed that 

 a comparison of the preceding theorems wdth the results of a 

 particular mode of intei-preting quaternions (explained in the 

 July Number) enabled us to account, « priori, for the connexion 

 observed by Mr. Cayley between certain quaternion formulae and 

 those which occur in the theory of the rotation of solids. (Phil. 

 Mag. Ser. 3. vol. xxxiii. p. 196.) 



In fact, since the quaternion cos 6+ sin 6{il+jm-\-hi) maybe 

 considered to represent the rotation of a radius vector through 

 an angle 6 round an axis whose direction cosines are /, m, n ; or, 

 which comes to the same thing, the description of a correspond- 

 ing arc on the sm-face of a sphere ; and since the descriptions 



