1892 - 93 .] Prof. Knott on Innovations in Vector Theory, 235 
(11.) Throughout his pamphlet, for reasons given in his letter to 
Nature, Gibbs refuses to admit that the complete product of two 
vectors has any claim upon his attention. He, nevertheless, 
smuggles it in, as we saw in § 9, so as to facilitate his treatment of 
the “ dyadic ” versor. If it is not really needed, then we must 
look upon it as a kind of catalytic agent. In general, however, the 
expression a/5 is taken to mean a “ dyad ” or operator of the form 
aSyS or /5Sa, according as the operand is affixed or prefixed. 
What, then, are we to understand by the following equations : — 
= jJ/AvVio (2) of § 1 64 
and 
Jdpm = JJda- X Vm (2) of § 165 
in quaternion symbolism, where by da is meant vds, v being unit 
vector drawn perpendicular to the surface element ds. Now, these 
equations are perfectly true in their ordinary quaternionic interpre- 
tation ; but if we are to credit Gibbs with consistency, we cannot 
believe that he means us to regard da-o), dpw, and V w as quantities 
(i.e., quaternions). They ought to be dj^ads. If we take them as 
prefactors, we simply reproduce Ko. I of the respective sets of 
equations, namely. 
JJ'ddU J'dpu x Vw 
But if we take them as postfactors, we get for the first case, 
JJ'iii^vrds =JJJh>r\j ,ijidv. 
an equation which is interpretable and true only if S V r = 0, — that 
is, for an incompressible fluid. Similarly, the second equation 
becomes 
f%Tdp=JfiTvV.w:U, 
an equation which is interpretable and true only if Sv V t 0. But, 
since equations (2) are meant to be perfectly general, we must con- 
