41G 



HITCHCOCK. 



are linear in x and are rational functions, of these points and which shall 

 contain the polynomial as a factor. 



One further illustration may be given of selecting the b's so that the 

 determinant shall not vanish identically, and of factoring the determi- 

 nant. Take N = 5, K — 2, and let the seven b's be [3145], [3356], 

 [3346], [a:567], [3467], [3457], and [3456]. The elements of the first 

 row of our determinant will be (3145/) (2367 'j) where j = 8, 9, • • •, 14; 

 and similarly for the other rows. This determinant does not vanish 

 identically. For if we make the substitutions as = ai + a 2 , ag = a2 

 + a 3 , aio = ai + a 3 , an = ai + a 7 , ai 2 = a 2 + a 7 , ai 3 = a 3 + a 7 , and 

 an = y, an arbitrary vector, the determinant becomes 3 



(31452) 

 (12367); 



; 



; 



; 



; 



; 



; 



; Pi ; 



(33562) (33561) 



(12473); (12473); 



(33462) (33461) 



(12573); (12573); 



P 2 

 



P 3 

 



Pa 



P 6 



; P 5 

 ; Pi 

 ; P 9 



o 



o 



o 

 o 



o 

 o 





 



; ; P 8 



(35671) (.r5672) (.t5673) 



(12347); (12347); (12347); P m 



(34671) 0r4672) (.x4673) 



(12357); (12357); (12357); P u 



(34571) (34572) (34573) 



(12367); (12367); (12367); P 12 



; ; ; P 



where Pi, • • • P12 denote elements which are of no consequence, but 

 Po = (3456#) (1237?/). It is evident that this determinant factors 

 into a number of linear determinants together with the two-row 

 determinant 



(33562); (33561) 



(33462); (33461) 



and the three-row determinant 



(35671) 

 (34671) 



(34571 ) 



(.r5672); (35673) 

 (34672); (.r4673) 

 (34572) ; (34573) 



(53) 



3 For compactness in printing this determinant, the two factors of each ele- 

 ment are set one above the other. Thus the element in the first row and first 

 column is 



(31452) (12367) 

 and similarly for the other factored elements. 



