eB + dB - 



e^ 



eB 



1- 



dB 



2-T + 



dB 





LdB = o 



(leB 



dB _ 



dB 



eB) 



= eB 



Macfarlane — Differentiation in the Quaternion Analysis. 203 



therefore, eB^dB _eB = eB [dB + ^^^" + 



( 2 ! 



/>13+ dR nR I A 



and 

 and 



Hence -— = eB. 



dB 



dgb 



In form the differential coefficient does not differ from that for 



do 



dB"^ 

 The next step is to find —-^ where B denotes a vector logarithm. 



Since /? 1 p ^' -B" 



eB = \+B+— + . . . + — + 

 2 ! nl 



and deB = eBdB, 



it follows that dB" = nB"'^dB. Hence, if B denote any vector 

 logarithm, real or imaginary, and n a positive integer, 



^§ = nB-^; and ^ = ^^""^ f , 



if t denote any scalar variable. 



The above symbol B denotes a vector in the Hamiltonian sense ; 



it is really an imaginary vector, and can be analysed into h J- 1/3, 

 where b denotes the magnitude, and /3 the axis. Now 



B^ = {bj^i^y = - v~fi' ; 



and this expression reduces to - h~ on the principle that /3^= 1. 

 According to Hamilton's assumption, 



dB-" _ d{- ¥) __^.dl^ 

 ~di ~ dt ~~ di' 



but, according to the results of the above investigation, 



dt dt ^ dt ' 



and this reduces to - 25 — only when fi is constant. The above 



