228 Proceedings of Roycd Society of Eclinhurgli. [sess. 
where each term is a dyad, is called a dyadic trinomial, or simply a 
dyadic. Any linear and vector function can he expressed as a 
dyadic. 
In his letter to Nature, Professor Gibbs argues strongly in favour of 
the dyadic as a functional symbol. To Hamilton’s Scr^p there corre- 
sponds (T.{a\ + Pix-\-yv).p, in which the dyadic may be considered 
to act either on p or cr. If on a-, then (i.{ak-^ j^ix + yv) corresponds 
to Hamilton’ s <j>'(T where the conjugate of <^. This conjugate 
may also be written (Xa + fx/3 + vy).o-. In this double representa- 
tion of the conjugate function, Professor Gibbs believes that his 
notation is much more flexible in analysis than Hamilton’s, and 
admits of a freer development than the notation aSA. + ^Sp-. -l-ySy. 
That may, or may not, be so ; but the question is not between the 
merits of an essentially artificial notation like aX -f fS/x -f yv and those 
of an expanded semi-cartesian form like aSAp + ^Spp + yBvp. The 
question is, whether the dyadic can do what the quaternion operator 
^ cannot do. 
As an example of the artificiality of the dyadic, take the defini- 
tion of the direct product * of two dyads (indicated by a dot). 
Here, by definition 
{a^} . {yS} =/?.yaS. 
Quaternions gives at once 
^i//'p = aS/?(ySSp) -f ifec. = aS8pS/?y -p &C. 
There then follow the definitions of the sJceiv products of and p, 
thus 
cfiXp — aXxp-\- (3p. X p -p &c. 
pX(]i = pxaX-\-px -p &c. 
These are operators or dyadics. To see what they mean, let them 
operate on some vector a. Then we find 
cj) X p.cr = aSXpa- fSSpLcr -h &c. = (fVpa- 
p X cji.cr = VpaSXa- -P Vp/ISpo- -p &C. = A'p^cr. 
* Gibbs calls the quantity (p. <r (which is simply Hamilton’s <pir) the direct 
product of the dyadic <p and the vector c. The direct product of two vectors 
is a./3( = - Sa/3), and this Heaviside calls the scalar product. Similarly trans- 
lating the Gibbsian dialect, he speaks of (pir as being the “scalar product of 
the dyadic and the vector ” — and gets a scalar product equal to a vector ! 
This “ is most tolerable and not to be endured.” Gibbs’s own use of direct and 
of its symbolic “dot” in tAvo cpiite different senses is itself open to criticism. 
