of the Linear Vector Function. 547 



(0) to that plane be (N), and (S) the centre of figure of the 

 triangle. With notation 



OR = r, ON = n, NReu, 0S = s, SR = k, . (3 a) 



the quoted proposition leads to 



V 



a + b + c 



= r = n4-uEs + k, .... (4) 



where on clearing of fractions the last members revert to 

 using the composition with one scale-factor (a + b-\-c) ; a vec- 

 torial addition is visualized without distortion in the graph. 

 To locate the intersection (R,), take (D) in (AB) satisfying 

 (AD/DB = &/a) ; the line (CD) contains (II), the ratio of 

 segments being (DR/RC = c/(a + fr). Hence with given 

 intercepts [x l9 y h z^) the direction of (Vj is fixed by two 

 ratios ; the magnitudes of their terms do not enter. Planes 

 parallel to (ABC) preserve these ratios, and only magnify 

 its graph by changing the sum (a + b-t-c). This segregation 

 is a working convenience. 



Next particularize on making (OA, OB, OC) the unit- 

 vectors themselves. Since here (a7 1 =^ 1 = ^ 1 = l), the scale- 

 factors (a, b, c) become numerically the tensors (V : , V 2 , V 3 ). 

 It is common practice that the latter shall include every 

 dimensional adjustment. Distinguish the special values 

 of (3 a) as (r', n', u', s', k'), and remark that (n' = s', u'—k') 

 because (N, S) now coincide. Choose (w' = io'wi) a t right- 

 angles with (u/) in the present plane (ABC) ; (n', u') are 

 perpendicular, and another correct general form for (V) 

 follows, provided that the tensor [id') is given proper magni- 

 tude and dimensions : 



V = (V 1 + V 2 + V 3 )(n' + u') = (V 1 + V 2 + V 3 )[n' + (n'xw')]. 



... (5) 



With the indicated order of factors (n', w'), accept the 

 separation in the second member as conforming more closely 

 to physical data. Then (w') is a merely mathematical 

 auxiliary, whose sense in its line must be governed by 

 either one convention or the other concerning circulation ; 

 (w') must be reversed at transition between right-handed 

 cycle and left-handed. But should the third member 

 register the primary record of physics through its variables 

 and its form, the alternatives regarding assumed circulation 

 clearly reverse (n'). As though, without changing the circu- 

 lation-rule, we came to deal with a new vector by applying 



2 N 2 



