230 Proceedings of Royal Society of Edinburgh. [sess. 
where m represents what unit volume becomes under the influence 
of the linear operator. The dyadic form of is 
{ySxy/xXv + yXayxX + aX ySA. X /x}, 
which Professor Gibbs further proposes (in Nature) to symbolise by 
Thus, burden after burden, in the form of new notation, is 
introduced apparently for the sole purpose of exercising the faculty 
of memory. Further on in the pamphlet the quantity m is intro- 
duced in the form and is called the determinant of the dyadic ; 
and then we are told that the relation of the surfaces VA'/x' and 
VA/x “ may be expressed by the equation ” 
VA>' = M^,-^VA/x. 
May he expressed ! — as if this concise quaternion representation were 
a mere notation, overshadowed by the effulgence of the beautiful 
line-long bracket of dots and crosses given above. 
(9.) On page 42 of Gibbs’s pamphlet we have a beautiful example of 
giving back with the left hand what has been sternly removed with 
the right. We read ; — “ On this account we may regard the dyad as 
the most general form of product of two vectors. We shall call it 
the indeterminate product.” And then he shows how to obtain a 
vector and a scalar “from a dyadic by insertion of the sign of skew 
or direct multiplication.” 
This is exquisite. From the operator aA -f /?/x -l- yi/ he forms — 
heedless of his high-toned scorn for anything like a quaternion pro- 
duct — the conception of the sum of three such products, but quiets 
his conscience by calling them indeterminate I This sum of pro- 
ducts then becomes, by simple insertion of dots and crosses, the 
vector 
= axA-f;Sx/x-fyXi/ 
and the scalar 
0,. = a.A -h ^./x -f y.y 
Language is impotent to characterise aright this remarkable feat 
of jugglery. “The quaternion product,” we are told in effect, “is 
an abomination. But here is a nice thing we call the dyad, a vector 
operator of the purest blood, which (if occasion serve) may neverthe- 
less be regarded as the most general [and therefore non-vectorial] 
form of product of two vectors. Being called indeterminate, it is 
