Macfarlane — Differentiation in the Quaternion Analysis. 207 

 In a similar manner it may be shown that 



-O 



The generalisation of Taylor's Theorem for such vectors is now 

 evident. Let X and Y denote any two vector logarithms, real or 

 imaginary; then 



/(X+ Y) =/(X)+/(X)r+l/'(X) r^- + etc, 



provided that the order of X prior to Yis> preserved throughout. And, 

 in general, such vector differentiation differs from the common diffe- 

 rentiation in preserving the real order of the vector symbols. 



Let R denote a variable vector of the kind which we have been 

 denoting by JB. Then R = rp, where r is the modulus, and p the 

 axis. 



dR = dr . p+ rdp, 



and d'^R = d^r . p + 2dr dp + rd'^p. 



Suppose that the vector is circular (or imaginary), then 

 p = v/' - 1 {cos ^ . «■ + sin ^ (cos ^ . / + sin ^ . Ic)], 

 where «', y, k are constant real axes. Then 



Hence, dR = drp + rdO r-r + rd(^ - , 



d'S = d'r . p + (2(ird6 + rd'B) ^ + {2ird4, + rd-tf,) ^ + r(,dey ^ 



,rm''^,*^'-ded^^- 



d^p 

 Now W^~^' 



and ^ = - sin^ (9 . p - sin ^ cos ^ . r^, 



and ::r7^ = cot d -— ; 



