QUATERNIONS IN THE ALGEBRA OF VECTORS.
169
we adopt is a matter of minor consequence. In order to keep within the resources of an ordinary printing office, I have used a dot and a cross, which are abeady associated with multiplication, but are not needed for ordinary multiplication, which is best denoted by the simple juxtaposition of the factors. I have no especial predilection for these particular signs. The use of the dot is indeed liable to the objection that it interferes with its use as a separatrix, or instead of a parenthesis.
If, then, I have written
and
for what is expressed in quaternions by
and
and in like manner
and
for
and
in quaternions, it is because the natural development of a vector analysis seemed to lead logically to some such notations. But I think that I can show that these notations have some substantial advantages over the quatemionic in point of convenience.
Any linear vector function of a variable vector
may be expressed in the form—
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where
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or in quaternions
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where
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If we take the scalar product of the vector
and another vector
we obtain the scalar quantity
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or in quaternions
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This is a function of
and of
and it is exactly the same kind of function of
that it is of
a symmetry which is not so clearly exhibited in the quaternionic notation as in the other. Moreover, we can write
for
This represents a vector which is a function of
viz., the function conjugate to
and
may be regarded as the product of this vector and
This is not so clearly indicated in the quaternionic notation, where it would be straining things a little to call
a vector.
The combinations
etc., used above, are distributive with regard to each of the two vectors, and may be regarded as a kind of product. If we wish to express everything in terms of
and
will appear as a sum of
each with a numerical coefficient. These nine coefficients may be arranged in a square, and constitute a matrix; and the study of the properties of expressions like
is identical with the study of ternary matrices. This expression of the matrix as a sum of products (which may be