310 PROCEEDINGS OF THE AMERICAN ACADEMY 



XXI. 



INVESTIGATIONS IN QUATERNIONS. 



THE THESIS OF A CANDIDATE FOR MATHEMATICAL HONORS CONFERRED 

 WITH THE DEGREE OF A.B., AT HARVARD COLLEGE, AT COMMENCE- 

 MENT, 1877. 



By Washington Irving Stringham. 



Presented by Professor Beivjamin Peirce, Jan. 9, 1878. 



I. Logarithms of Quaternions. 



1°. Hamilton proves (Lectures, p. 54G) the following theorem: 

 The tensor of the sum of any number of quaternions cannot exceed 

 the sum of their tensors. 



If p and q be any two quaternions, then 



T{p + qy = {p + q){Kp + Kq) = Ty + Tq' + 2S.pKq 



= Tp2 4- Tq^ + 2TpTqS\J.pKq 



= (Tp + Tqy - 2TpTq(l - SV.pKq) ; 



and since the scalar of a versor cannot fall outside the limits ± 1, 



... T(;, + q) -<:Tp + Tq. 



The same proof holds for any number of quaternions, and gives in 

 general 



T(;> + ? + r + . . . ) -<T;, + T^ _|_ Tr + . . . 



On the foregoing theorem depends the test of convergence of the 

 series 



Let -Z„ represent the first n terms of .^; S = the series formed by 

 the tensors of ^', /S„= the first n terms of S. The terms of aS are 

 represented by the general form — = a„. Each successive term of 



