114 Mr. F. L. Hitchcock on the 



ordinary algebraic principles if the order of vectors in any 

 term is not changed. The expressions thus obtainable with 

 nabla are noteworthy, both for their simplicity and their 

 limitless variety ; a few are given below, but are to be 

 regarded as exemplifying the method rather than as consti- 

 tuting any attempt at completeness. 



In the second place, as a vector, nabla satisfies the formulae 

 proved by Sir W. R. Hamilton for vectors and quaternions, 

 so that by using these we may change the order of multipliers 

 and bring nabla next to any desired operand. The variety 

 of results obtainable by this method is even greater than by 

 the other. The transformations which take the place of the 

 commutative law of ordinary algebra are wonderfully many- 

 sided in their application. 



2. As a significant example of these two ways of working 

 with nabla, we may take the problem of applying nabla to 

 the product of any two functions. Call these functions 

 q and r. They may be either scalars or vectors ; in general 

 they are quaternions. They are assumed to be finite and 

 continuous in the portion of space considered. We then 

 have immediately, by ordinary differentiation, 



V( f ) = VVr + vV, .... (1 A) 

 where, on the left, nabla acts on both q and r, but on the 

 right is distributed, as indicated by the accents, according to 

 the familiar rule for the differentiation of a product ; that is, 

 in the first term on the right, nabla acts on q but not on r, 

 and in the last term it acts on r but not on q ; the order of 

 multiplication is not changed in either term. 



The first term on the right may, by the associative law of 

 multiplication, be rewritten as V<7 • r ; this gives 



VO) = Vfl'-r + W* .... (IB) 

 a convenient equation for some purposes (to which I shall 

 recur below). If, now, we wish to bring V> m the last term, 

 next to the operand r, we may do so by means of the formula 

 K( M )=K 9 K ?I ..... (2) 

 true for any two quaternions q x and q *. For this, by 

 operating on both sides with K, becomes 



M = K(K S rKy 1 ), (3) 



because KK = 1 f. This formula shows that we may invert 

 the order of any two quaternions by introducing the symbol K. 

 With regard to notation, the parentheses on the right may 

 well be omitted, for the first K obviously applies to all that 

 follows it, otherwise the two adjacent K's would together be 



* Hamilton, ' Elements of Quaternions,' 2nd ed., Art. 192. 

 f Ibid. Art. 145. 



