:10 v. 9 f 



124 Dr. L. Silberstein : Further Contributions 



and draw from T a a tangent to k. Then the contact point /> 

 will be the end-point of the required unit vector b normal 

 to a. The proof that T b a will then also be a tangent may 

 be left to the reader. Again, the other tangent from T a will 

 touch re in b', which will be found to be collinear with 

 6, 0, so that V = — b. Similarly, the other tangent from T b 

 will touch k in a the end-point of —a. Thus, together 

 with Oa, Ob also Oa, Ob' and Oa' , Ob will become auto- 

 matically pairs of orthogonal rays. In short, the straights 

 aol and bV will be orthogonal. Since T a is the pole of bb\ 

 -and T b that of act', we see that our orthogonal lines are, in 

 usual nomenclature, pairs of conjugate lines with respect 

 to k. At the same time Ave see that, in technical language, 

 0T a T b will be a self-polar triangle with respect to k. 



We thus know how to draw to any straight passing 

 through a perpendicular in 0. This limitation to the 

 origin can be easily removed. In fact, let us generalize our 

 original definition by postulating that if (ab) = 0, and if v be 

 any vector equal to b, then also 



(av)=0; 



more generally, if a' = a and b' = b, then (aV) = (ab). Now, 

 till vectors equal to a are coterminal with a. Thus, returning 

 to fig. 3, all lines of the pencil centred at T h will be ortho- 

 gonal to all those of the pencil centred at T a . In particular, 

 the tangent to k at any of its points (as T b a) will be normal 

 to the corresponding diameter [a a) *. 



This shows us also that the proper representatives of the 

 orthogonality thus defined are the pairs of corresponding 

 pencil-centres themselves, each pair being so correlated as 

 T a , T b in fig. 3. (These pairs of points are, in common 

 nomenclature, " conjugate " with respect to the conic k.) 

 We may have an opportunity to return to similar questions. 

 Meanwhile let us proceed with our chief subject. 



4. Scalar Product denned. Distributivity. Let A= f A a 

 and B— | B b be any two vectors. 



Let us denote by (AB), and call the scalar product o£ A 

 into B, the number 



(AB)= A | . IB (ab) (6j 



Since the tensors are ordinary numbers and (ab) = (baj. 

 we have also 



(AB) = (BA). 



the commutative property. 



* Notice in passing that the T-line itself will be orthogonal to every 

 other straight line whatever. 



