414 HITCHCOCK. 



a,(1234) - ai(j234) + a 2 0'134) - a 3 0124) + a 4 (jl23) = (41) 



and by multiplying by [xa 5 a ; ] 



(x5jl) (J234) - (x5j2) 0134) + (.r5j3) (J124) - (.r5j4) (J123) = (42) 



which, being an identity, holds when j runs from 6 to 10. Hence 

 when a 5 is common to all four b's the four equations are linearly 

 related. 



When the four b's do not contain an a in common the four equa- 

 tions are independent. For suppose the b's to be [xaia 5 ], [xa 2 a 5 ], 

 [xa 3 a 5 ], and [xaia 2 ]. The four corresponding equations are 2(.rl5j) 

 (234j)C'j = where j runs from 6 to 10, and three other equations of 

 similar form. The determinant of the coefficients of the first four 

 columns is 



(*156) (2346), (.rl57) (2347), (.rl58) (2348), (a-159) (2349) 



(a-256) (1346), (z257) (1347), (z258) (1348), (.r259) (1349) 



(a-356) (1246), (.r357) (1247), (.r358) (1248), (a-359) (1249) 



(a-126) (3456), (.rl27) (3457), (.rl28) (3458), (.rl29) (3459) 



(43) 



In order that this determinant might vanish identically it would be 

 necessary and sufficient that four numbers C\, c 2 , c 3 , c 4 exist, independ- 

 ent of a, and not all zero, such that 



Cl (xl5j) (234j) + c 2 (x25j) (134;) + c 3 C.r35j) (124;) + Ci (xl2j) (345j) 



= (44) 



for the vectors a 6 , a 7 , as, a 9 , represented by a,-, are arbitrary. If we 

 let &]■ = x + a 5 this equation reduced to its last term, namely 



c 4 (.rl25) (345a:) = 



whence c 4 = 0, for the vectors x, ai • ■ • a 5 may have any values what- 

 ever. By letting a, be ai -f- a^, a 2 + a 4 , and a 3 + a 4 we see that ci, 

 c 2 , and c 3 must all vanish, contrary to hypothesis. Hence (43) does 

 not vanish identically. We may proceed similarly whenever the four 

 b's do not have an a in common. 



This determinant must accordingly contain as a factor Cw, a quad- 

 ric through the nine points ai,- • • a 9 . That it vanishes at all these 

 points can be verified by inspection : if x = a 4 the last two rows are 

 equal numerically and opposite in sign; if x equals any other a all 

 the elements of a row or of a column vanish. 



This determinant is of the fourth degree in x and in ar • ■ a 5 . It is 

 of the second degree in &$■ ■ • a 9 . Since the factor C\q is a quadric in 



