142 Further Contributions to Non-Metrical Vector Algebra. 



these vectors will also be associative (P.V.A., p. §), that is 

 to say, (X + Y) + Z = X + (Y + Z). Thus all addition rules 

 will continue to hold, together with their consequences, 

 among which is the construction of the standtian scale, and 

 so on. Again, " conies " can be generated on cr, as loci of 

 crosses of correlated lines of two projective pencils, say by 

 the process 



E=[X + T1*, 



as explained in P.V.A., Section 15. One such closed conic k 

 can be drawn round and used as a standard conic, and 

 the whole line of reasoning given above can be literally 

 repeated, leading to the definition of orthogonal lines and 

 thence to the concept of the scalar product of two vectors. 

 This product will again obey the distributive law so that, for 

 instance, the generalized theorem of Pythagoras, C 2 = A 2 + B 2 , 

 will continue to hold for " right-angled " spherical triangles, 

 and so on. In short, we shall have an ordinary vector 

 algebra, or a parabolic measurement system, set up upon the 

 contemplated portion of the spherical surface. On the same 

 surface we can also set up a hyperbolic system (by taking 

 one of the said real " conies " as Cayley's "absolute" and 

 adopting his definition of distance and of angle) or an elliptic 

 system of measurement, which latter will simply be the 

 usual spherical trigonometry. In the last two enses we are 

 deprived of the facilities of an associative and distributive 

 vector algebra, while in the first case we have a vector algebra 

 consisting of addition and scalar multiplication formally 

 identical with ordinary vector algebra. 



The same remarks and explanations apply, mutatis mutandis, 

 to the case of three-dimensional space, where also the vector 

 product of vectors comes to its rights. 



9. Supplementary Constructions to Section 5. — It has seemed 

 advisable to add a few words about the constructions involved 

 in what was treated in Section 5, and in particular on p. 132. 



A. Given an angle a= (a, b) =aOb, say ivith the centre of k 

 as vertex, to construct an angle a! = <x with any given 0' as 

 vertex and I as one of its sides, coplanar with a, b. 



Let T L be the terminus of /. Draw the join T lt a cutting k 

 again in c. Double the angle a in the explained way, 

 making aOd = 2ot. Join cd cutting the T-line in T . Then 

 0' T will be the required second side of «' = a. In fact, the 



angle acd is 2(~) =a; the termini of the sides of this angle 



are T t T : hence, T l O r T' = a.' = a. 



