56 



spherical opening of that pyramid ; or the area of the spheri- 

 cal polygon, of which the corners are the points where the vec- 

 tors a, |3, 7, . . K, X, meet the spheric surface described about 

 their common origin with a radius equal to unity. And by 

 combining this result with the general method stated to the 

 Academy by the Author* in November, 1844, for connecting 

 quaternions with rotations, it is easy to conclude that if a 

 solid body be made to revolve in succession round any 

 number of diiferent axes, all passing through one fixed point, 

 so as first to bring a line a into coincidence with a line j3, by 

 a rotation round an axis perpendicular to both ; secondly, to 

 bring the line |3 into coincidence with a line y, by turning 

 round an axis to which both j3 and y are perpendicular ; and 

 so on, till, after bringing the line k to the position X, the 

 line X is brought to the position a with which we began ; then 

 the body will be brought, by this succession of rotations, into 

 the same final position as if it had revolved round the first or 

 last position of the line a, as an axis, through an angle of 

 finite rotation, which has the same numerical measure as the 

 spherical opening of the pyramid (a, (5, y, . . k, X) whose 

 edges are the successive positions of that line. 



* The same connexion between the Author's principles, and geometrical 

 or algebi-aical questions, respecting the rotation of a solid body, or respect- 

 ino- the closely connected subject of the transformation of rectangtdar co- 

 ordinates, was independently perceived'T)y Mr. Cayley ; who inserted a com- 

 munication on the subject in the Philosophical Magazine for February, 1845, 

 under the title, " Results respecting Quaternions." It is impossible for the 

 Author, in the present sketch, to do more than refer here to Mr. Cayley's 

 important researches respecting the Dynamics of Rotation, published in the 

 Cambridge and Dublin Mathematical Journal. An account of the speculations 

 and results of the late Professor Mac Cullagh on this subject may be found 

 in part viii. of the Proceedings of the Royal Irish Academy ; and a summary 

 of the views and discoveries of Poinsot has been given by that able author 

 in his very interesting tract, entitled, Theorie Nouvelle de la Rotation des Corps, 

 Paris, 1834. 



