of the Linear Vector Function. 555 



So long as complete coincidence of (S 1? S 2 ) is excluded, the 

 last expression cannot vanish. It is indeterminately satis- 

 fled for {n } =n 2 ; /°2 = co )> which at once justifies the 

 proposition and adds its necessarv condition. It is to 

 inspection true that a plane parallel to any difference- 

 diagonal cuts off equally proportional segments of both 

 components, and of their half-sum. In illustration, inter- 

 pret what comes of combining equations (14, 25) and 

 dividing by (a 2 + 6 2 + c 2) : 



:p=(S + k 2 )-(s4-U 2 ' f ). 



a 2 + b 2 + c-2 a 2 + b 2 + c 2 a. 2 -f b 2 + c 2 



. . . (35) 



The vector (p) will be both perpendicular to (s) and parallel 

 to (ABO) ; (U 2 , Q 2 ") neutralize each other's components 

 in (s). If (u 2 ") completes a rectangular decomposition of 

 (k 2 j in (ABC), it will lie in (SN) of (3a); but (p, u 2 ") 

 may be oblique. Of course the second member is valid 

 throughout two groups of vectors, subject to a proviso of 

 common difference between pairs. And again there is 

 brought forward a reminder of equation (18). 



Return to equations (27, 29, 30), whose parts (d 2 , Ct 2 ") are' 

 less special than those supposed in equation (35). When 

 (a 2 + b 2 \ c 2 ), (a 2 + b 2 + c 2 ) are unequal, a common divisor 

 does not ''reduce (Q 2 , Ct 2 ") to the plane (ABC)"; some 

 analysis of the latter relations is neededi Let the projec- 

 tions of (r 2 , r 2 ") have tensors (x 2 , t/ 2 , z 2 ), (x 2 n , y 2 \ z 2 "). 

 Equating equivalents for each of (Q, 2 , Cl 2 ;/ ), record the 

 consequences 



(a 2 4- b 2 + c 2 )x 2 = a 2 x x ; (a 2 + bj + cj)x 2 " = a 2 x u etc. ; \ 

 m 2 _ a 2 + b 2 -rc 2 a 2 x 2 " b 2 ?/ 2 " _ c 2 z 2 " 



m 2 " ' a 2 ! + b 2 ' + c 2 ' a 2 'x 2 b 2 y 2 c 2 z 2 



_ a 2 x 2 " + b 2 y 2 " -f c 2 z 2 " m 



. (36) 



a 2 'x 2 -\-b 2 'y 2 + c 2 'z 



2 



whose second and last members connect interchange of 

 scale-factors with the ratio (m 2 /m 2 "). Because change of 

 cycle cannot neglect it, that crossed pairing of intercepts 

 (r 2 , r 2 ;/ ) and scale-factors (m 2 , m 2 ") should be examined in 

 some detail, and its bearing upon conjugate relation extracted. 

 To this end, begin with two general vectors (S l5 S 2 ) drawn 



