of the Linear Vector Function. 549 



outlined under equations (5, 6, 8), appear as 



Vi =|(V 1 + V 2 + V 3 )[v 1 -f(i^ 3 - w )v 1 ] f 

 Vi"=KVi+V. + Vt)[Ti-(^i»*-i»kiflOvil ; ■ (11) 



etc., etc. 



The insertion of the unit-tensors (i^, u 2j v z) preserves the 

 formal type ready for algebraic evaluation. These pre- 

 liminaries have broad lv suggestive aspects : the conversions 

 of (n') into (V, V', V") are after the pattern of stretch- 

 operators and quaternions, for instance. But equations 

 (5) to (9) also place a patent emphasis upon taking due 

 account of axial combinations with the operand-vector, if 

 (V, V', V") become generally representative of linear vector 

 functions and of relations among them. This is for the 

 present our chief concern. 



' Return accordingly to equation (4) and repeat the above 

 sequence, modified only as that more general form demands. 

 Begin with the arbitrary partitions, which may finally be 

 algebraic sums, 



a = ai-\-a 2 ; A = 6 1 +5 2 ; c = Ci + c 2 ; . . (12) 

 and express (V) to match their terms : 



V = [(«!#!>, + (%l> 2 + (^i)Vs] 



+ [(a 2 #i)vi + (%i)V2 + (c 2 yi)v s ] = Clx + U 2 . (13) 



The components (ft 1? Ct 2 ) are given by means of the same 

 intercepts ; their graphs utilize the same plane (ABC), 

 though scale-factors are varied. Make equation (4) model 

 and put 



Qj = (a x + h + Ci)^ = (a 2 + 6i + CiXn + Uj) = (a t + b x 4- c\) (s + k t ) 

 a 2 = (a 2 -+- b 2 + c 2 )r 2 = (a 2 + 6 2 + c 2 ) (n + u 2 ) = (a 2 + ?> 2 4 c 2 ) (s + k 2 ) 



The arbitrary character of (Q. l9 Q> 2 ) enters their geometry 

 through the elements (u l5 u 2 , k 1? k 2 ), since (n, s) belong to 

 both. We can treat (r, r 2 , r 2 ) as making real intersections 

 with the plane (ABO), whose equation their extremities 

 satisfy, which yields useful corollaries. The recomposition 

 of (di, Ct 2 ), by applying to the second members the general 

 construction detailed specially for equations (3, 4), is simple 

 to verify. The third members reproduce the scheme of 



:} d4) 



