318 PROCEEDINGS OF THE AMERICAN ACADEMY 



9°. The angle of a quaternion has not the usual generality of an 

 angle in trigonometry; it is never negative or greater than 180*^. 

 (19) must, then, be modified as follows, to give the principal logarithm. 

 It is easily seen that if 



(^Z?)> 2mQ and <(2w +1)0, 



and that, if 



(^Z?)> (•2m + 1)0 and <(2m + 2)0, 



Znq= (2m + 2)0 — I'Zq. 



Hence, the principal logarithm of Tlq becomes 



inq = y ]Tq + (l^Zq — 2m0)UVy, (20) 



if {^Zq)> 2»iQ and <{2m + 1)0 ; 

 or 



\nq = :^\Tq-\- [^Zy — (2m + 2)0]UV5', (20)' 



if {^Zq)> (2m + 1)0 and <(2m + 2)0. 



For the power of a quaternion, these formulae become 



1^" = nlTq J^{nZq — 2m0)UVy, (21) 



if (nZq)> 2m and <(2m + 1)0 ; 



or I?" = nlTq + [7iZq — (2m + 2)0]UVy, (21)' 



if (nZq)> (2m + 1)0. and <(2m + 2)0. 



For the principal logarithm of any integral power of a vector : — 



1«4 n _ inlTa, 



1^4 « + 1 _ (4,j _|_ i)iT« + ^aUa, 



l„4n + 2 _ (4,j^ 2)lTa ± QJJa* 



1^4n + 3 — (4„ _|_ 3)iT« — iQVa. 



The cases in which Iq" = 7i\q are those in which {7tZq)<Q- 

 For ]q» we shall always have 



Iq^^hTq + TZq.VYq, 

 * But for Ua in this formula, any unit-vector may be substituted. 



