1892-93.] Prof. Knott on Innovations in Vector Theory. 225 
V 2Pot Zi = + irru * 
or Potw= 47rV"“Zi 
Similarly, if w be a vector quantity, 
Pot O) = 47T V ““CO 
Then we have 
jSTew'zz= V Pot w = 47 t V ~ 
Lap CO = W Pot CO = 47 tV V “^co 
-M ax co= SVPot(o = 47rSV-''co 
Thus (Lap - Max) co is 47 t V ~^co, which probably Gibbs would write 
Tai CO — only, alas ! he cannot use it.t 
Kow, Gibbs gives a good many equations — theorems, I suppose, 
they ape at being — which connect those functions and their various 
derivatives. All these equations are, in quaternions, identities, 
involving the very simi:)lest transformations. But there is no such 
flexibility and simplicity in Gibbs’s analysis. For example, he takes 
eight distinct steps to prove that 
V V “co = CO, 
and even then he does not get it in this perfectly general form. He 
has to prove the special forms which this identity assumes, according 
as (0 is solenoidal or irrotational. He finds that “ -—Pot and V x V 
47T 
are inverse operators ” for solenoidal functions, and that “ -—Pot 
47T 
and - W. are inverse operators ” for irrotational functions — all of 
which is included in the above identity. For if a) = coj + CO 2 , so that 
SV(Oi = 0 and VVco 2 = 0, we get 
CL>i + (O 2 = 0) = V “ V (Y’^ci)]^ + SVCO 2 ). 
Another of the theorems given, namely, 
47t Pot (0 = Lap Lap co - Kew Max o> 
Tait proves this well-known equation of Poisson by purely quaternionic 
methods. It is an interesting commentary on the “simplicity” of tlie 
positive sign — the fetish alike of Heaviside and Macfarlane — that the quater- 
nion gives 4-r positive on the right-hand side ! 
t Taking his cue from Gibbs, Heaviside might possibly find Ham a less 
“ludicrously inefficient” name than Habla for the operator v. 
21/1/93. 
VOL. XIX. 
P 
