120 Dr. L. Silberstein : Further Contributions 



the properties of (ab) from this somewhat implicit definition, 

 will be shown presently. 



Meanwhile, in order to be able to treat vectors with any 

 origin other than 0, let us recall from P.V.A. (section 8) 

 the definition according to which all equal vectors are, in 

 the first place, co-terminal, i. e. concurrent with one another 

 in a 7 -point. Thus, if T a be the terminus of a= Oct, and it 

 is required to construct, with any given 0' as origin (not on 

 T), the vector equal to a, we have first of all to draw 0' T a ; 

 next, to obtain its end-point, we have to draw 00' crossing 

 the T-line in T', say ; then the straight aT' w T ill cross 0' T a 

 in a\ the end-point of the required vector. Having thus 

 obtained, on the line 0' T a , the unit, we can in the well- 

 known way, construct upon it the vectors equal to 2a, 3a, 

 and so on, in general 0'A' = A'=\A' a. This enables us to 

 speak in a perfectly definite sense of the tensor of a vector 

 drawn anywhere on the contemplated plane, as of the 

 number of equal projective (unit) steps contained in it. 

 Similarly we can construct, from 0', vectors equal to 

 "b= 06, c=0c, etc., and thus reproduce (not to say 

 " transfer ") the original standard k at any spot where it is 

 desired *. Notice that if k! be such a reproduction of the 

 standard round 0', the original T-line will again be the 

 polar of 0' with respect to k' . We thus see that our original 

 is by no means a privileged point of the contemplated 

 two-space. Any 0' is as good a point. The 2-line, how- 

 ever, does play a privileged role, being the common polar of 

 the " centres " of all these standard conies with respect to 

 each of them. It is also the polar of 0' with respect to any 

 «/, the homologue of k . 



But it will "be remembered that even the T-line, although fixed for 

 the duration of an investigation, is but an arbitrary reference line, and 

 can, in different instances, be chosen in a manner most suitable for the 

 investigation of the given figure or figures drawn on the plane, the only 

 geometrical entities. How the choice of this reference line influences 

 the position of the " centres " of the standard conies k we already know ; 

 each of them is the pole of T with respect to k'. 



* The physicist, will notice the difference between the standard k 

 and, say, the metre standard preserved at Paris. It is not enough to 

 make a copy of the latter at the spot, in Paris, it has to be carried 

 about; the former, however, is "reproduced'' or constructed at the place 

 where it is just required. This is a capital difference. Moreover, it is 

 enough to construct a few isolated points of *'. For, as is well known, 

 five points of <' would (even in the absence of T) suffice to construct, 

 without further appeal to k, any number of other points of k . 



