Colloquium publications.

32

THE CAMBRIDGE COLLOQUIUM.

diameters V1 and V2, W1 and W2 will be conjugate semidiameters
in an ellipse similar to this and similarly placed.
22. Summary of the Properties of the Linear Vector Function. For the
purpose of expressing the condition (2) in convenient form, it is desirable to
have at hand a few of the important formulae for the manipulation of linear
vector functions. They are for convenience summarized in this section.
A vector W is a linear vector function of a vector V if when V = c1Vl + c2V2,
where Cl and c2 are constants, it follows that W = c1W1 + c2W2. The properties of this relation can be briefly and conveniently expressed in terms of the
Gibbs concept of dyadic.
A dyadic is the sum of symbolic products of the form
A = a1 + a2l2+ + " * + ~ak3k,
the nature of the product being defined merely by the fact that when the
dyadic is combined with a vector p on the right (notation A * p) a new vector p'
is produced, which is given by the equation
(3) p' = A-p = al(3-p) + a2(2'Pp) + * ' + ak(3k'p).
Obviously, according to this equation, the vector p' is a linear vector function
of the vector p. Two dyadics are said to be equal if they represent the same
transformation, that is, if when combined right-handedly with an arbitrary p,
the same vector p' in both cases is produced.
Similarly we can define multiplication on the left of a dyadic by a vector,
and show that the resulting vector is uniquely determined, that is, that two
equal dyadics produce the same vector by multiplication on the left, although
of course generally a different vector from that produced by multiplication
on the right.
The following properties of dyadics follow immediately from the definition:
(a) p'. (A.p) = (p'.A).p,
so that the notation p'.A -p has no ambiguity.
(b) a(3 + y) = a + ay.
(c) If
A1 = a '#I' + a2'fl2' + * C* + ak'ek,
A2 = al"/l + a 2- ** + an''' n
and As is the dyadic such that A3-p = A2 (A1 p), then it may be expressed
in the form
(4) As = A2.A1 = i"i '"(i"aj')f,
whence it is called the product of A1 by A2. In general A2 A1 is not the same
as A1.A2.
(d) If A3 is the dyadic such that A3sp = A1.p + A2-p it may be expressed
in the form
As = Xi c'3i' + 2 j oa'i"',
whence it is called the sum of Al and A2. Obviously:
Al + A2 = A2 + A1.