Double Nature of Nabla* 123 



Fy as a function of p ; or it is the product of / 2 and/ 2 . We 

 may then, as in the last paragraph, arrive at a sum of terms 

 of the form r\/'sp't, but by (4 B) 



Vs = *V-2VVsV, 



whence we bring V next to p and have a sum of expressions 

 of the form 



rsVp .t — 2rVYsV .p . t, 



and because V/>=— 3 and W«V.p = — 2Vs*, we obtain a 

 sum of expressions of the form 



— 3rst+4:rYs .t, 



and may write in general 



/V.Fy = S(-3r«« + 4rV«.0 . . . (30) 



in which r, s, and t are the quaternions obtained by expanding 

 the right side of (28) or (29). The result (30), apart from 

 practical utility, has a certain interest by showing (what is 

 well known), that we may find the nabla of any quantity 

 (and the more general /V), without ever introducing axes of 

 reference, whether rectangular or not ; and by means of one 

 differentiation, not three, — a fact which might be commended 

 to the attention of those who think that quaternions are mere 

 abbreviations for Cartesian expressions. 



We may,. however, if we choose, introduce i, j, and k into 

 the work. The original definition of V gives, by (29), 



/V • Fry = fi .f 3 i +fj .f 3 j +/* ./A . . . (31) 



and the right side is an invariant of the two linear functions 

 / and / 3 . The late C. J. Joly showed how such invariants 

 may be calculated, given the functions / and / 3 f. 



A fourth method, sometimes the shortest in practice, is to 

 write the expression/ V'-./sp' as a sum of terms in whatever 

 way may be most convenient, then calculate the invariants 

 of each term separately, and add. The advantage of this 

 method is that it is always possible to express a linear function 

 as the sum of a small number of standard forms, whose 

 invariants are easily remembered; as, for example, that 

 used above, VY>'V . p = — 2 Vs, or the very common form 

 V73Sap' = -ap. 



To sum up, we may find the result of applying any linear 

 function of nabla to any given function by performing a 



* Tait, ' Quaternions,' Art. ]45-6. 



f Appendix to Hamilton's ' Elements/ 2nd ed.. chapter v. ; and Trans. 

 K.I. A. xxx. p. 709. 



