

(37) 



556 Graphical Synthesis of the Linear Vector Function. 



from (0), and let (R 1? R 2 ) be given by their inversion ; so 

 that in notation like equation (34), 



1 Bi 1 = n^ i ; R 2 = ?iiS 2 ; ^(R 1 + 'R 2 ) = np 1 ' ; ' 



The mean scale-factor (n) is common to " (Sx + S 2 ) , (Ri -i- Rg) ? 

 but (p u pi) differ in general. Omitting steps of plain 

 reduction, the working out shows 



i(Si - S 2 ) f i(R x - R 2 ) = KSi + Ei) - i(S 2 + R 2 ) = n( Sl 

 f (Si + S 2 ) - i(Bi + »*) - i(Si - »i) + i(S, - R 2 ) 



i(S 1 -S,)-i(Bi-B a ) = i(S 1 -E 1 )-i(S s 



n T — n 2 

 2 



B.) 



™i 



(si-s 2 ) ; 



^(si + s,)^ 



. . . (38) 



It is almost intuitive that the mean factor (n) should corre- 

 spond to the mean vectors in the way that these equations 

 confirm. The second and third are interpretable at sight in 

 their last members as expressing colinear vectors parallel to 

 the plane (ABC). Denote these respectively by (2D l5 2D 2 ) ; 

 it is clear that a changed sign for (?i 2 ) interchanges them. 

 In terms thus chosen, it follows that 



S 1 (-D 1 + D 2 ) = R 2 + (D 1 + D S ); S 2 + (D 1 -D 3 ) = R 1 - (D, - D 3 ) n 



(S 2 -R 2 -2D 2 ) + (S 1 -R 1 -2D 2 ) = 0; 

 (S 3 + R 2 + 2D 1 )-(S 1 H-R 1 -2D 1 ) = 0. J 



. . . (39) 



If next the extra condition be attached that (s 1 ~s 2 ) shall 

 be perpendicular to (s), the symmetries of equations (39), in 

 comparison with those of equations (29, 30, 31), make corre- 

 spondence permissible of (M), either with (2D X ) or with (2D 2 ), 

 according to the actual occurrence of the positive sign for 

 the scale-factors. In this scheme, equation (35) is easy to 

 locate as a special combination ; and equation (19) finds its 

 place. The last of equations (39) bring to light the well- 

 known " symmetrical average function." The important 

 part played in all these developments by the mean factor (n) 

 must be explicitly recognized. That is intimately associated 



