of the Linear Vector Function. 551 



as projections into the radius vector linking with equa- 

 tion (4), 



# 1 v 1 +y 1 v 2 + ZiV 3 =3s. .... (21) 



This diagonal from (0) o£ the parallelopiped (#i?/i~i), deter- 

 mined by given axis-intercepts, takes prominent place among 

 the data. The second factor in each vector product is seen 

 to specify legitimately a vector parallel to a coordinate plane, 

 (Y 1? Z l5 Z 2 , X 2 , X 3 , Y 3 ) being adjusted positive magnitudes. 

 Because (Ct 2 ) is arbitrary, the third members must be inde- 

 pendent ; which bars their general union into one vector 

 product of whicji (3s) is a factor. Observe that the specifi- 

 cations parallel to the coordinate axes in this group amount 

 in the aggregate to 



(X 2 +X g )v 1 + (Y 1 + Y 3 )v 2 + (Z 1 + Z 2 )v 3 =A. . (22) 



Further, since every such summation by planes effectively 

 takes each component by axes twice, the last requirement 

 can be met as a total with 



,A= X ^v 1+ ^v 2+ ^v 3 , . (23) 



assigning components to axes. On the face of it, this sub- 

 stitutes an average, and effaces some particulars of equation 

 (20). The latter can be summed into these terms : 



a 2 =[3sxA]-[(y 1 Z 2 -0 1 Y 3 )v 1 



+ (^X3-^ 1 Z 1 )v 2 +0r 1 Y 1 -y 1 X 2 )v 3 ] ; . (24) 



an important result that we put into the condensed notation, 



a 2 =M+a 2 '; a 2 + a 2 '=M; a 2 =M+a 2 "; a 2 '+a 2 "=o ; 



. . . (25) 



where the double sign has been made to allow (M, Q 2 ) as 

 alternates for the resultant diagonal. Evidently (&/) admits 

 a new sum of vector products : 



<V - [(yiv 2 + *iv 8 ) x ( Y s v 2 + Z 2 v 3 ) ] 



+ [OiV 3 + SjvO x (ZiV, + X 3 v0] 



+ [(^iV 1 +^ 1 v 2 )x(X 2 v 1 + Y 1 v 2 )]. . . (26) 



The first factors throughout (Q, 2 , Ct 2 ') are identical, and the 

 average fixed by equation (23) reappears; though in general 

 the distributions differ, since the second factors of (Ct 2 , Gt 2 ') 

 contain pairs that are complementary within (A). Con- 

 tinuing to coordinate the graphs through the plane (ABC) ? 



