1910-11.] The Less Common Special Forms of Determinants. 321 
a + a b + a c + a d — b a — c a ~d a 
-a b b + b c + b d + b a - c b - d b 
— a c b c c + c d + c a + c b d c 
— a d -b d — c d d + d a + d b + d c 
= abcd+'£abc(d a + d b + d c )+'2 J ab(c d d a + .... ) +'£a(b c c d d a + ....). 
The arrangement of the two developments almost raises doubts as to whether 
the “ rule ” had been utilised, suggesting indeed that in the latter instance, 
for example, the cofactor of ab was first obtained in the form 
C d + Ca + ^ ~ d c 
-c d d a + d b + d c , 
and the cofactor of a in the form of a similar determinant of the third 
order. The “rule,” however, is noted by Cayley in Crelles Journal , lii. 
(1855), p. 279. 
The number of terms is (n-\- 1)" -1 , n being the order-number of the deter- 
minant. This Sylvester obtains by putting a, a b , a c , . . . all equal to 1. It 
will be observed that from the form of the development we have 
1+3-2 + 3-3 = 4 2 
1 + 4-3 + 6-8 + 4-16 = 5 3 
1 + 5-4 + 10-15 + 10-50 + 5*125 = 6 4 
Borchardt, C. W. (1859, May). 
[Ueber eine der Interpolation entsprechende Darstellung der Elimina- 
tions-Resultante. Crelles Journ., lvii. pp. 111-121 : or Monatsb. 
d. Akad. d. TFiss. (Berlin), pp. 376-388 : also abstract in Annali di 
Mat., ii. pp. 262-264.] 
The representation in question is in terms of the values which the two 
functions <p(x) and both of the n th degree, assume for the values 
a 0 , oq, a 2 , . . . , a n of x. It emerges as a special determinant of the form 
-l-(ll) 
-(12) .. 
. . -(1») 
-(21) 
-(2m) 
-(»!) 
-(«2) •• 
. . <r n - (nn) 
where or r = ('r0)-l-( , rl)-f .... +(rri) and (rs) = (sr ), — a form which we 
readily recognise to be the axisymmetric case of Sylvester’s determinant of 
the year 1855. To the consideration of it Borchardt, supposing it to be new, 
devotes the last six pages of his paper. 
VOL. XXXI. 
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