102 MULTIPLE ALGEBRA. 



where rj, o> are the distances of the plane from the four points, 

 and x y y, z, w are the coordinates of any point in the plane. Here 

 we may set 



p^gx+ijy + gz + cow, (15) 



and say that p represents the plane. To some extent we can intro- 

 duce this letter into equations instead of r\, w. Thus the equation 



lp'+mp"+np'" = (16) 



(which denotes that the planes p', p", p'", meet in a common line, 

 making angles of which the sines are proportional to I, m, and ri) is 

 equivalent to the four equations 



Jf+mf + rcf" = 0, Zi/ + mjf + 7ij/" = 0, etc. (17) 



Again, we may regard r\, f, w as the coordinates of a plane. The 

 equation of a point will then be 



xg+yr) + z+wco = Q. (18) 



If we set 



q = xg+yr) + z+ww, (19) 



we may say that q represents the point. The equation 



, 

 ^ ' 



2 ' 



which indicates that the point q'" bisects the line between q' and q", 

 is equivalent to the four equations 



" + '" n"4-n'" 



f-^T*-, tf- 1 ^, 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 Mobius 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 



> 



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



m'+mir+ir"=o (23) 



has precisely the same meaning as equation (16) of ordinary algebra- 

 That the point Q lies in the plane II is expressed by equating to zero 

 the product of Q and II which is called by Grassmann external and 

 which might be defined as the distance of the point from the plane. 



We may write this 



QxII = 0. (24) 



