550 Prof. F. Slate on a Graphical Synthesis 



equation (5), and justify as general expressions 

 Q>i = («i + b x + ci) [n + (n x w x )] 



(15) 

 d 2 = (« 2 + b 2 + c 2 ) [n + (n x w 2 )] ; J 



where (w l5 w 2 ) successively replace (w'). Obviously, the 

 ideas underlying equations (8, 9) continue into this phase, 

 which enables us to indicate compactly a conversion of (n) 

 into any complementary constituents of (V). And when a 

 mean vector (w) has been determined from 



(a + b + c)w=( ai + b 1 + ci)w! + (a 2 + b 2 + c 2 )w 2 ; u = n x w ; 



. . . (16) 

 (Qi, Q 2 ) can be summed into 



V = (a + 5 + c)[n + (nxw)]. . . . (17) 



But an attempt at similar general conversion of (s) into 

 (Q*i, Q 2 , V) halts at the possible obliquity to (s) of 

 (k 1? k 2 , (k)). The situation that involves mean vectors (s, w) 

 to this extent dislocated, remains for analysis after taking 

 out three particular cases. First, the mean vector (w) may 

 prove to vanish ; (V, n) become colinear. Or (s) can be 

 perpendicular to (k), though oblique to each separate part. 

 Or again, (s) can be perpendicular to both (k 1? k 2 ), yet not 

 coincident with (n). If for the then colinear sum the 

 relation among tensors be added : 



k 1 (a 1 + b 1 + c 1 ) + k 2 (a2 + b2 + C2) = k(a + b + c) ; and k = sxq, 



. . . (18) 



with permissibly oblique factors, the type of equation (17) 

 is retained by 



V = a l + Q 2 = (a + & + c)[s+(sxq)], - . (19) 



whose peculiar lesson is more fully read later. 



Under natural guidance of several aspects in the foregoing- 

 results, reserve one component (say (&i)) as stated for equa- 

 tion (13), and transform (Q 2 ) for general discussion through 

 the equivalents 



(a 2 x l )v l = (3/!V 2 + ^v 3 ) x (Y^ + Z^) = (yJL Y - eiYi)vj n 



(%i) v 2 = (*iY 8 + flriYi) X (Z 2 v 3 + X 2Vl ) = ( 2l X 3 - ^iZ 2 )v 2 ; I (20) 



(c 2 1 )v 3 =(f 1 v 1 +yiv,) x (XsVj + Y 8 v s ) = (a; 1 Y 8 — yiX,)v 3 .J 



The vector products employ first factors that are distances 

 from the lines of (v b v 2 , v 3 ), and that combine significantly 



