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( x i ~ Vi )~ 2 (* i -2/ 2 )' 2 (* i -2/ s)" 2 (*1 ~ Vi )^ 
(* 2 ~ 2 / i Y "' (*2 - y 2 )' 2 (* 2 - 2 / 3 )‘ 2 fe - 2 / 4)' 2 
(* 3 - 2 / l )' 2 fe -^ s )' 2 (*3 -%)' 2 fe - 2 / 4)" 2 ' ’ 
(* 4 - 2 / i )‘ 2 (* 4 - 2 / 2 )' 2 (* 4 - 2 / s )' 2 (* 4 - 2/ 4 )' 2 
the four other determinants on the right being equal to zero. For 
example, the first of the four 
(*i-yi)" 1 ( a «-2/i)' 1 <*k - »*)-*(** - 
(*2 2 / l ) '(*1 “ 2 / i ) 1 (* 2 - 2 / 2 ) 1 (* 1 - 2 / 2 ) 
(* 3 - 2 / l )" 1 (* 2 - 2 / l )' 1 (*3 ~ 2/ 2 ) -1 (*2 — 2 / s ) 1 
(* 4 - y 1 ) _ 1 (* 4 - J / l )' 1 (* 4 - 2 / 2 )' 1 (* 4 - 2/ 2 )" 1 
from which, if we remove the factor 
(*1 - 2/ i) _1 (*2 - 2/ i) _1 (*3 - 2/ i) _1 (*4 - 2 / 1)' 1 
X (*1 - 2/ 2 ) -1 (*2 - 2/ 2 )' 1 (*3 - 2/ 2 )' 1 (*4 - 2 / 2)" 1 
X (*1 - 2 / 3) _1 (*2 - 2/ 3 ) -1 (*3 - 2/ 3 ) _I (*4 - 2 / s )' 1 
X (*1 - 2/ 4 ) _1 (*2 - sO ' K *! - 2 / 4) _1 (*4 - 2 / 4)" 1 
there remains the cofactor 
(*2 “ 2 / l )(^4 — 2 / 1 ) (^2 “ V 2 }{ X 4 : ” 2 ^) 
( x s ” 2 h )(^4 “ 2 / l ) (^3 — 2 / 2 ) (*^4 “ 2 / 2 ) 
(^1 - 2 /i) 0 *h “ 2 h) (^1 “ 2/2) (*^4 “2/2) 
« ® • • * 
Diminishing each element of the first row in this by the corre- 
sponding element of the second row, and each element of the 
second by the corresponding element of the third, we can remove 
the factors and have resulting a determinant with 
its first two rows identical. 
(/8) is the well-known theorem of Borchardt* regarding double 
alternants. 
10. The case of (a) for the 3rd order, which was first given by 
Cayley, t is worthy of a little special attention, on account of a 
* 
Cr die's Journal , liii . p . 194 . 
t Ibid., lvi . p . 184 . 
