432 Prof. Donkin 07i the Geometrical Laws of the 



product has been since noticed both by Mr. Cayley (Phil. 

 Mag. vol. xxxiii. p. 196) and Mr. Boole (vol. xxxiii. p. 278); 

 I have introduced the above proof of it partly for the sake of 

 future reference, and partly because I am not certain whether 

 Mr. Boole refers to it as a proposition depending upon Mr. 

 Cayley's results, or as the analytical expression of an indepen- 

 dent geometrical theorem, under which latter aspect I have 

 here viewed it. 



I shall conclude this paper by showing how the formulae of 

 M. Rodrigues may be established by means of quaternions. 

 The investigation is closely connected with the subject of the 

 preceding observations, and it will be seen to include an ex- 

 planation of another remarkable analytical result noticed by 

 Mr. Cayley (Phil. Mag. vol. xxvi. p. 142). 



Recurring to fig. 1, and recollecting that the radius of the 

 sphere is represented by unity, let /, ?«, n be the coordinates 

 of the positive pole of AB, and .r, t/, z the coordinates of the 

 positive pole of AC, both referred to the fixed axes, with which 

 the moveable axes at first coincide. Also let 



2AB = 5, 2AC = (p. 



When the moveable sphere has been displaced by the ro- 

 tation AE=fl, let 



a,b,c; a',b>,c'; a", b", c" 



be respectively the direction cosines of the moved axes, referred 

 to the fixed axes. The plane of the great circle (on the move- 

 able sphere) which at first coincided with that of AC, now 

 coincides with that of DE, so that ^, j/, z are also the coordi- 

 nates of the positive pole of DE referred to the moved axes. 

 Hence if ^, )j, ^ be the coordinates of this same pole referred 

 to ihej^xed axes, we have, by the common formulae, 



^=ax + a'^ + a"z 



yi=ibx + Vy + Wz 



}^=:.cx + dy-\-d'z. 

 Now let 



5-= cos- + sin - {il+Jm + kfi) 

 g' = cos ~ + sin ^ {ix -\-jy + kz) 



<7"= cos I + sin|(f^+j>j + ^0' 



Then, applying the calculus of quaternions to the triangles 

 ABC, BDE, it is easily seen that we have 



