152 Mr Brill, On Quaternion Functions, with especial [Feb. 23, 



out to the end, as the inverse of a quaternion function would not 

 in general correspond with the quaternion function framed on the 

 model of the inverse scalar function. Still further difficulties 

 would arise if we attempted to apply the scalar notation to 

 functions of two quaternions. 



So far as I am aware, Boole 1 was the first to give a general 

 expression for a function of a single quaternion framed on the 

 model of a specified scalar function. His expression may be easily 

 deduced from Sylvester's Interpolation Formula 2 in the Theory of 



Matrices, which states that if \, \, \ be the latent roots of 



an w-ary matrix, then 



where m denotes the matrix, provided that none of the latent 

 roots are equal. 



The identical equation satisfied by a quaternion is 



q*-2qSq+(Tqy = Q, 



and, therefore, the latent roots of the quaternion are given by the 

 equation 



X i -2\Sq + (Tqf=0; 



and since (Tq) 2 = (Sq) 2 + (TVqf, 



it is clear that the roots are 



Sq + cTVq and Sq-iTVq. 



These roots are obviously distinct except in the case when q 

 reduces to a scalar. Thus we have 



1 {f(Sq + iTVq)+f(Sq-LTVq)} 



2 



+ I UVq {/(Sq + cTVq) -f(Sq - cTVq)}, 



Ail/ 



which is Boole's result. 



1 " On the Solution of the Equation of Continuity of an Incompressible Fluid," 

 Proc. E. I. A., vi. 375—385. 



2 This theorem was stated by Sylvester in a paper in the Phil. Mag. for 

 October 1883, entitled " On the Equation to the Secular Inequalities in the 

 Planetary Theory." It also appears as the second law in his "Laws of Motion 

 in the World of Universal Algebra." 



