(46) 



ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 415 



all ten vectors the other factor must be a quadric in x and in ai • • • a 5 

 and be independent of a 6 - • a 9 . If we wish to find this latter factor 

 it is allowable to assign to a 6 - • a 9 any values we please. If we take 



a 6 = a 3 + a 4 , a 7 = a 2 + a 4 , and a 8 = a 2 + a 3 (45) 



the determinant (43) becomes 



(.rl59) (2349) 



, (a-254) (1342), (z253) (1342), (a-259) (1349) 



(a-354) (1243), , (a-352) (1243), (a-359) (1249) 



, (a-124) (3452), (.rl23) (3452), (*129) (3459) 



which is equal to the product of (x354) (1243) (.rl59) (2349) times the 

 determinant 



(x254) (1342), (a-253) (1342) 

 (zl24) (3452), (*123) (3452) 



which in turn is 



(1342) (3452) [(a-254) (zl23) - (xl24) (.r253)]; (48) 



but from (41), by letting a, = x and multiplying by [xaoa 5 ] 



- (z251) (a-234) - (x253) (xl24) + (*254) (.rl23) = (49) 



that is, the quantity in brackets in (48) is equal to (.r251) (.r234). 

 Collecting results, we see that the determinant (43), by virtue of the 

 substitutions (45), become the product of eight determinants: 



(.r354) (a-251) (a-234) (zl59) (1243) (1342) (3452) (2349) (50) 



It is evident by inspection of this product that the adventitious factor 

 of the determinant (43) must be 



(.t354) (a-251) (1234) (51) 



for in no other way can we pick from (50) a factor quadratic in ar • • a 5 

 and x. The algebraic sign is, however, undetermined. 



Determinants similar in form to (43) can evidently be written down 

 for any values of K and N. The second factor of each element will in 

 every case be a polynomial of degree K — 1 in the variables which 

 enter into it. The first factor is linear in x and in N — 1 other points. 

 We have therefore the theorem : 



Theorem V. Given a 'polynomial homogeneous of degree K in N 

 variables, and n — 1 points at which this polynomial vanishes: it is in 

 general possible to write down a determinant of order q whose elements 



