546 Prof. F. Slate on a Graphical Synthesis 



functions that is in some respects supplementary to the 

 traditional methods and more direct*. 



Let (0) be a base-point for three vectors (V 1? V 2 , V 3 ). 

 Suppose them physically homogeneous (compoundable) and 

 mutually perpendicular. Their union to a resultant (V) 

 drawn from (0) is shown fundamentally in terms of unit- 

 vectors and their tensors by 



VvEEV^ + V^ + VgVs (1) 



The unit-vectors (v l5 v 2 , v 3 ) in this order of their indices 

 may fit either cycle — right-handed or left-handed. Tensors 

 cannot be negative under a primary ruling that every vector 

 shall be rated positive in its own "forward " direction. This 

 agrees, moreover, with algebraic usage regarding the radius 

 vector of polar coordinates, and the perpendicular from the 

 origin to any plane. Negative tensors are attendant later 

 upon the algebra of assembling vector elements parallel to 

 the same line, after giving them a unit-vector in common. 

 The reversed (neutralizing) vector ( — V) can be described 

 equally well both ways : 



(-V)v=(~Y 1 )v 1 + (-Y 2 )v 2 + (-Y 3 )v 3 n 



W=V 1 v 1 ' + V,v/+V,y,' ; V. . (2) 



v + v' = v 1 -fv/ = v 2 + v 2 / = V3 + v 3 ' = 0. ) 



Note that the cycle (circulation-rule for axial vectors), with 

 the same index-order, must 'exchange between right-handed 

 and left-handed at passage from (v 1? v 2 , v 3 ) to (v/. v 2 ', v 3 ') 

 or vice versa. Represent (V 1? V 2 , V 3 ) graphically by the 

 arbitrary lengths (OAe^V!, OB =3/^2, 00 = ^^) asso- 

 ciated with the scale-factors (a, b, c) such that 



Xl) : 3 ;}(3 } 



V = (aa?i) v x + (%i)v 2 + (c£!)v 8 



and consequently, ■— V= (a^v/ + (6y 1 )v 2 ' + {cz^v. 



As a standard, all six factors of (v 1? v 2 , v 3 ) will be positive ; 

 but consistent algebra covers the usual variations. Let (V) 

 cut the plane of (ABC) at (R), the foot of the normal from 



* The generalized construction belongs to the older developments for 

 centre of inertia. Its essential thought is carried back to Leibnitz by 

 Minchin : 'Statics,' vol. i. p. 18 (1884). It 'is given bare recognition in 

 recent books, but is scarcely made of any important service. See Greaves. 

 ' Elementary Statics/ p. 19 (1886) ; Love, ' Theoretical Mechanics,' p. 15 

 (1897). Its control over scale-factors adapts it well to "contraction 

 hypotheses," adding a resource there. 



