410 PROCEEDINGS OF THE AMERICAN ACADEMY. 



We may extend the idea of complements .to scalars and pseudo- 

 scalars. The complement of the scalar n will be defined as the pseudo- 

 scalar nku; the complement of the pseudo-scalar iils-u will be defined 

 as the scalar — n. This may be written 



(«ki4)* = ?iki4'ki4 = — w, 



thus establishing the convention kn^ku = — 1. It may readily be 

 shown that, for any two singular vectors p and q of different class, 

 the outer product is the complement of the inner product, that is, 



pxq = (p.q)ki4. 



In other words the inner and outer products of singular vectors are 

 numerically equal. 



Some Differential Relations. 



16. As the inner product r-r of a vector by itself is numerically 

 equal to the square of the interval of the vector r, the equation of 

 the unit pseudo-circle of which the radii are all (7)-lines is r-r = 1; 

 and the equation of the conjugate unit pseudo-circle of which the 

 radii are (5)-lines is r-r = — 1. As the tangents to a pseudo-circle 

 are perpendicular to the radii, they must be of opposite class. A 

 pseudo-circle of which any tangent is a (5)-line (the radii being (7)- 

 lines) is called a (6) -pseudo-circle; and a pseudo-circle of which 

 any tangent is a (7)-line (the radii being (5)-lines) is called a (7)-pseudo- 

 circle. In general if a curve has tangents which are all of the same 

 class (5) or (7), the curve may be designated as a (5)- or a (7)-curve; 

 the normals to the curve will then be respectively of the opposite 

 class (7) or (5). The interval of the arc of any such curve will be the 

 limit of the sum of the intervals of the infinitesimal chords along the 

 arc. We shall not be obliged to consider any curve which is not 

 altogether of one class as here defined. 



As dr is the infinitesimal chord as a vector quantity, the formula 

 for the scalar arc is 



= I V(/r.r/r or .s' = / 



V- dT'dr 



according as the curve is a (7)- or a (5)-curve. 



The sectorial area in a unit pseudo-circle may be regarded as the 

 sum of infinitesimal right triangles, of which the area is numerically 

 equal to ^rxc/r if r is drawn from the center. The numerical 



