to Non-Metrical Vector Algebra. 121 



3. Geometrical meaning of (ab). Definition of Orthogo- 

 nality. — In order to perceive clearly the intrinsic nature of: 

 the ordinary number (ab) or (ba) as a property of the vector 

 pair, a, b, let us construct graphically some vector expression 

 into which (ab) will enter alongside with things already 

 known to us. For this purpose let us return to the equation 

 (1) of the standard conic k. From this we find easily, as 

 the equation of the tangent to k at any of its points x, J/, 



[a?+.(ab)y]*+[y + (ab)a?]y=l, 



./', y being the coordinates of any point of the tangent line. 

 In particular, the tangents at the end-points a(#= L, y = 0) 

 and bQ/ = l, # = 0) of the vectors a, b used as axes will have 

 the equations 



« + (a% = l, (O 



(ab>+y = l (t b ) 



Let P be the cross of the two tangents t a , t b , or the pole 

 ■of the chord ab as polar. Then its coordinates, the solutions 

 of the last two equations, will be 



,. 1 



^ _7? ~l+(ab)' 



and the vector P= OP, which is ^a + 7;b, will become 

 S 



l+(ab) 



where S = a -hb. . . . (2) 



It will be remembered that S, the end-point of S, is the cross 

 of aT b with bT a (P.V.A., 4). 



This is the required expression, or equation, since it 

 contains, besides (ab), only P and S, well determined and 

 easily constructible things. Hitherto (ab) was defined only 

 as the coefficient of 2acy in the coordinate equation of tc with 

 a, b as axes. Henceforth we can drop that equation and 

 consider (ab) as defined by the vector equation (2), which 

 shows at once the intrinsic nature of (ab) . Such being the 

 case we can, on the other hand, expand (ab) using, for 

 instance, the components of a and b along some auxiliary 

 axes ; for we shall then know that the value thus obtained 

 is independent of these auxiliaries. We may profit from 

 this possibility later on. 



Meanwhile let us concentrate our attention upon the last 

 equation. It will teach us many interesting things. First 

 ■of all, 1 -f (ab) being a mere number, it shows us that <S is 



