55 



or rather the more concise but equivalent expressions (35), 

 have been seen to offer themselves as coefficients of a symbolic 

 equation of the third degree (38), which is satisfied by a cer- 

 tain characteristic of operation a, connected with the solution 

 of a certain other symbolic but linear equation : and the 

 Author may be permitted to mention that this is only a par- 

 ticular (though an important) application of a general method, 

 which he has for a considerable time past possessed, for the 

 solution of those linear equations to which the Calculus of 

 Quaternions conducts. To those who have perused the fore- 

 going sections of this Abstract, and who have also read with 

 attention the Abstract of his communication of July, 1846, 

 published in the Proceedings of that date, he conceives that it 

 will be evident th-dtfor anyjixed poiiit A of any solid body (or 

 rigid system), there can be found (indeed in more ways than 

 one) a pair of other points B and C, which are likewise fixed 

 in the body, and are such that the square-root of the moment 

 of inertia round any axis AD is geometrically constructed or 

 represented by the line BD, if the points A and D be at equal 

 distances from C. 



VII. Finally, he desires to mention here one other theorem 

 respecting rotation, which is indeed more of a geometrical than 

 of a physical character, and to which his own methods have 

 led him. By employing certain general principles, respecting 

 powers and roots, and respecting differentials and integrals of 

 Quaternions, he finds that for any system or set of diverging 

 vectors, a, j3, 7, . . k. A, the continued product of the square 

 roots of their successive quotients may be expressed under the 

 following form : 



where 5 is a scalar which represents the spherical excess of the 

 pyramidal angle formed by the diverging vectors ; or the 



