Double Nature of Nabla. 115 



equal to unity and would not be written down. Thus we have 



?1? = KK ? K Sl (3 A) 



We may now write V in place of q l in this formula, for V 

 satisfies ad quaternion transformations. Accordingly 



Vq = KKryKV 



= -KKcyV, W 



because the conjugate of a vector is its negative * This 

 result is true whether V acts on q or not, since it depends on 

 the vectorial character of V and has nothing to do with 

 differentiation. On substituting in (1 B) we may drop 

 accents, with the usual understanding that an operator does 

 not act to the left of itself unless accents are inserted to that 

 effect; then V(?r) = Vff.r-KKjV.r, . . . (1C) 

 the period in the last term indicating that the first K acts 

 only as far as V. 



3. Operators such as KK#V, occurring in the last term 

 of (1 C), are not always entirely comprehended at first sight, 

 on account of their generality in comparison with ordinary 

 differentiation. A point of view from which all such 

 operators may be directly worked with is indicated below 

 (Art. 5) . It will suffice here to give three illustrations of the 

 methods by which we may, if we wish, introduce more 

 elementary operators to suit different purposes. 



First, suppose we wish to obtain, instead of KK^V, a form 

 which shall exhibit as obviously as possible its own relation 

 to ordinary differentiation. We shall then naturally bear in 

 mind that the notation connecting nabla with ordinary 

 differentiation along a direction in space is SoV, where a is 

 any vector along the direction of differentiation. In the 

 present case this vector will be Vq ; hence we shall so arrange 

 the work as to bring in a term of the form SV<?V. Thus 



W = VS? + W?, because S + V = 1, 



^Sg-.V+KVgV, by (3), 



= S?.V+(2S-l)V 7 y, because K=S-V=2S-1, 

 = (Sq-Vq)V + 2SVqV, identically, 



= Iv;.7-f lSV 7 V (4 A) 



and by substituting in (1 B), 



V(?r) = V/y • r+Kq . W -f 2SV?V . r, . (1 D) 

 a useful expansion which is in the form we started to obtain ; 

 * Ibid. Art. 144. 



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