to JV on- Metrical Vector Algebra. 123- 



vanishes, and 



(aa')=--l. 



These values correspond to what may be called the " zero angle/' 

 and the "'straight angle," respectively,— using these names only as- 

 synonyms of the pairs (a, a) and (a, a'). For hitherto we have not 

 introduced the general concept of " angle" as a magnitude. 



But the most important thing for our present purpose is 

 to note that when 



P = S = a + b. 



that is to say, when the pole of ah coincides with the cross of 

 aT h and bT a , our equation gives 



(ab) = (3) 



I propose to call such, manifestly remarkable, pairs a, b 

 normal or orthogonal, or perpendicular vectors. Their 

 definition can be conveniently written 



P a6 = a + b, (4) 



the double suffix reminding us that P is the pole of the 

 chord ab. It will be noticed that, when used as coordinate 

 axes, any such pair a, b reduces the standard-conic equation' 



t0 tl ' 2 +/=l. . (5> 



Fia\ 3. 



Any number of such orthogonal pairs can at once be con- 

 structed. In fact, from the form (4) of their definition we 

 see that both TJb and T b a are tangents to tc. Thus, if any 

 a= Oa (fig. 3) be given, prolong it up to its terminus '/', 



