548 Prof. F. Slate on a Graphical Synthesis 



the same scale-factor to ( + u'), 



V" = (V 1 + V 2 + V 3 )[n'-(n'xw')]=(V 1 + V 2 + V3)(n'-u'), 



... (6) 



convertible into (V) by imposing the new convention. The 

 last member involves a different intersection with the same 

 (ABC), marked by 



r'W + n"= Vi+ ^ + v> [u'+u» = 0]. . (7) 



This conical symmetry of (r, y") relative to an axis (n') 

 secures equal tensors (V, V"). Such opposed behaviour of 

 linear and axial vectors towards change of cycle is inherent 

 in separable assignments of axes and of their cycle-order. 



To illustrate how the above discrimination affects mixed 

 conditions, subdivide (w') into a mathematical auxiliary 

 (w") and a physical vector (w). Let (I/, A') express the 

 physical facts about linear and axial contributions to (V)„ 

 additive under its rule ; so that 



V=L / + A / = (Y 1 + y 2 + V 3 )[n^ + (n'xw")] 



+ [(V 1 +V 2 + V 3 )(Vxw)]. . (8) 



A change to the other rule, on whatever grounds executed, 

 is to be offset in (I/) through reversal of (w") as a detail of 

 mathematical routine ; the determinate physical element (w), 

 can only persist in direction, making thus the last term 

 negative. Then having 



V' = L'-A'; V-2A' = V; i('V + V')=L' ; i(V-V') = A'; 



. . . (?) 



(V, V') are mutually convertible on the basis explained for 

 (V, V /; ). Their sum is purely linear, and their difference 

 purely axial. Whatever is superficially ambiguous in really 

 occurring cases must be removed by exploration of phe- 

 nomena, in order to disentangle (w", w) . 



Adopt (v : /3, v 2 /3, v 3 /3) for coordinates of (N), and resolve 

 (w') of equation (5) as shown by 



w' = iv 1 v 1 + w 2 v 2 -\~w 2 v 3 (10) 



The suppositions include perpendicularity of (V, w'), which 

 implies one negative tensor or more in the last equation. 

 The group (Vi, V 2 , V 3 ) and corresponding components of 

 (V"), written in parallel expansion on either background 



I 



