Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/118

where $\xi ,\eta ,\zeta ,\omega$ are the distances of the plane from the four points, and $x,y,z,w$ are the coordinates of any point in the plane. Here we may set

$p=\xi x+\eta y+\zeta z+\omega w,$

(15)

and say that $p$ represents the plane. To some extent we can introduce this letter into equations instead of $\xi ,\eta ,\zeta ,\omega$ Thus the equation

$lp'+mp''+np'''=0$

(16)

(which denotes that the planes $p',p'',p''',$ meet in a common line, making angles of which the sines are proportional to $l,m,$ and $n$) is equivalent to the four equations

$l\xi '+m\xi ''+n\xi '''=0,$⁠$l\eta '+m\eta ''+n\eta '''=0,$⁠etc.

(17)

Again, we may regard $\xi ,\eta ,\zeta ,\omega$ as the coordinates of a plane. The equation of a point will then be

$x\xi +y\eta +z\zeta +w\omega =0.$

(18)

If we set

$q=x\xi +y\eta +z\zeta +w\omega ,$

(19)

we may say that $q$ represents the point. The equation

$q'''={\frac {q'+q''}{2}},$

(20)

which indicates that the point $q'''$ bisects the line between $q'$ and $q'',$ is equivalent to the four equations

$\xi '={\frac {\xi ''+\xi '''}{2}},$⁠$\eta '={\frac {\eta ''+\eta '''}{2}},$⁠etc.

(21)

To express that the point $q$ lies in the plane $p$ does not seem easy, without going back to the use of coordinates.

The form of multiple algebra which is to be compared to this is the geometrical algebra of Möbius and Grassmann, in which points without reference to any origin are represented by single letters, say by Italic capitals, and planes may also be represented by single letters, say by Greek capitals. An equation like

$Q'''={\frac {Q'+Q''}{2}},$

(22)

has exactly the same meaning as equation (20) of ordinary algebra. So

$l\Pi '+m\Pi ''+n\Pi '''=0$

(23)

has precisely the same meaning as equation (16) of ordinary algebra. That the point $Q$ lies in the plane $\Pi$ is expressed by equating to zero the product of $Q$ and $\Pi$ which is called by Grassmann external and which might be defined as the distance of the point from the plane. We may write this

$Q\times \Pi =0.$

(24)