Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/102

With these definitions, a great part of the laws of vector may be applied at once to bivector expressions. But an equation which is impossible in vector analysis may be possible in bivector analysis, and in general the number of roots of an equation, or of the values of a function, will be different according as we recognise, or do not recognize, imaginary values.

3. Def.—Two bivectors, or two biscalars, are said to be conjugate, when their real parts are the same, and their imaginary parts difler in sign, and in sign only.

Hence, the product of the conjugates of any number of bivectors and biscalars is the conjugate of the product of the bivectors and biscalars. This is true of any kind of product.

The products of a vector and its conjugate are as follows:

$[\mu +\iota \nu ].[\mu -\iota \nu ]$	$=\mu .\mu +\nu .\nu$
$[\mu +\iota \nu ]\times [\mu -\iota \nu ]$	$=2\iota \nu \times \mu$
$[\mu +\iota \nu ][\mu -\iota \nu ]$	$=\{\mu \mu +\nu \nu \}+\iota \{\nu \mu -\mu \nu \}.$

Hence, if $\mu$ and $\nu$ represent the real and imaginary parts of a bivector, the values of

$\mu .\mu +\nu .\nu ,$⁠$\mu \times \nu ,$⁠$\mu \mu +\nu \nu ,$⁠$\nu \mu -\mu \nu ,$

are not affected by multiplying the bivector by a biscalar of the form $a+\iota b,$ in which $a^{2}+b^{2}=1,$ say a cyclic scalar. Thus, if we set

$\mu '+\iota \nu '=(a+\iota b)[\mu +\iota \nu ],$

we shall have

$\mu '-\iota \nu '=(a-\iota b)[\mu -\iota \nu ],$

and

$[\mu '+\iota \nu '].[\mu '-\iota \nu ']=[\mu +\iota \nu ].[\mu -\iota \nu ].$

That is,

$\mu '.\mu '+\nu '.\nu '=\mu .\mu +\nu .\nu ;$

and so in the other cases.

4. Def.—In biscalar analysis, the product of a biscalar and its conjugate is a positive scalar. The positive square root of this scalar is called the modulus of the biscalar. In bivector analysis, the direct product of a bivector and its conjugate is, as seen above, a positive scalar. The positive square root of this scalar may be called the modulus of the bivector. When this modulus vanishes, the bivector vanishes, and only in this case. If the bivector is multiplied by a biscalar, its modulus is multiplied by the modulus of the biscalar. The conjugate of a (real) vector is the vector itself, and the modulus of the vector is the same as its magnitude.

5. Def.—If between two vectors, $\alpha$ and $\beta ,$ there subsists a relation of the form

$\alpha =n\beta ,$