57 
1913-14.] The Theory of Bigradients from 1859 to 18.80. 
Gunther’s proof is unnecessarily lengthy. The determinant can be readily 
transformed into one whose diagonal has for its elements 1 repeated m + 1 
times and m + 2 repeated m times, and whose other terms all vanish. For 
example, when m is 2, the requisite operations are 
row 5 - row 4 + row 2 , 
row 4 - row 3 + row 4 , 
row 3 — row 2 — rowq . 
Mansion, P. (1879). 
[On the equality of Sylvester’s and Cauchy’s eliminants. Messenger of 
Math., ix. pp. 60-63.] 
Mansion’s proof is not essentially different from the process of applying 
Trudi’s condensation-theorem to Sylvester’s bigradient. The additional 
fact, to which Mansion draws attention, namely, that many minors of the 
one eliminant have equivalents among the minors of the other, is also 
virtually included in Trudi. Thus, the four identities which Mansion 
indicates in the form (see his fig. 11) 
(Zq cq a 2 a B a 4 cl^ a,^ . 
G t>2 b B 
where the X’s, /Ps, Ps stand for 
&i a 0 b 2 + a 1 b 1 af) B + a l b 2 + afb 4 af> B + a 2 b 2 + « 3 & 4 - a 4 b 0 a 2 b s + a B b 2 - a b b 0 a B b B - 
b 2 a 0 b B + af) 2 a Y b B + a 2 b 2 a 2 b B + a B b 2 + a,b 0 a B b. 3 - a 6 b 0 + a 4 b 2 a b h x a 4 b B - a Q b l 
h a A a A a A ~ a 6 h o a A - a A a A~ a 6 b 2 
are only four of the ten noted by Trudi, the others being excluded, so to 
speak, by drawing three vertical dotted lines on the right of each determi- 
*0 
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a 4 
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a i 
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b 0 b x 
b 3 
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V 1 v 2 v 3 
V 4 
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ne 
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