118 Mr. ¥. L. Hitchcock on the 



We have 



dq 2 = dq.q + qdq 



= 2dq.q-2YYdqYq, 



by putting r = dq in (3 B), so that by (8) 



V</ = 2\7 f q'q-2VVVq'Vq, 



which is the same as (6). The method, we repeat, consists 

 in differentiating the given function, arranging the differential 

 in any way that may suit our purpose, and then replacing 

 dq by an accented q', while V is written at the left of each 

 term. 



To obtain the important generalization of (6) where the 

 exponent shall be any constant scalar whatever, we have 

 therefore to differentiate q n . To do so, we may think of the 

 differentiation as conducted in two steps: first, as if the plane 

 of q were fixed, and UVg constant, second, as if the rest of 

 the quaternion were constant. The sum of the two results 

 will be the complete differential *. Call the first step d and 

 the second d u then in general 



d = d + d l (9) 



The result of d on q n will be the same in form as if q were 

 a scalar, since quaternions of constant plane act like scalar s ; 

 that is 



d q n = d q . nq*- 1 (10) 



The result of d 1 implies no differentiation except dJJYq y 

 for q n is coplanar with q itself, VYq n is the same as dzUVg, 

 and we may write t 



q"=Sq n ±TYq n .TJYq, 

 whence, differentiating as if JJYq alone were variable, 



d iq n = ±TYq n .dJJYq, .... (11) 

 and by adding (10) and (1L) 



dq n = d q . 7iq n ~ ] ±TYq n . dUYq, 

 but by (9) 



d q = dq — d x q 



= dq-TYq.dJJYq (12) 



by the definition of <r/ l5 which substituted in the preceding 

 gives 



dq" = dq . nq n ~ l + dVYq . ( + TV^-TY^ . nq n ~ l ), . (13) 



* Hamilton, Art. 329. 



t The double sign is of course ambiguous in appearance only. We may, 



if we prefer, write + — |L instead of +T\ T q n in equations (11)-(14). 



