33 
1913-14.] The Theory of Bigradients from 1859 to 1880. 
In the section on the properties of the resultant (pp. 68-81) he recalls 
Richelot’s theorem of 1840, that if w be a common root of the equations 
a^x m + apc m ~ x + • • • = 0 , b^x n + \x n ~ x + • • * = 0 
whose resultant is R, then we have 
0R , 
0R . 
. ?R 
da Q ' 
0a x 
* da m 
0R 
. 3R . 
0R 
3 V 
1 06 x : ' 
' ' '' M n 
This is not brought forward as a theorem in determinants, hut for com- 
parison, when n = m, with Jacobi’s theorem of 1835 to the effect that the 
■signed primary minors associated with the elements of any row of Bezout’s 
condensed eliminant 
■are proportional to 
a o b i 
^0^2 I ! ®o^3 I "t" I ^l b 2 
m 
w m x , w m 2 , . . . , IV , 1 . 
In the section on common roots (pp. 81-84) he obtains such a root when 
it is solitary by taking any one of these three series of proportionals and 
dividing one member of the series by the member immediately following. 
When the number of such roots is lc he has recourse to the Sturmian 
remainders previously found, stating for comparison Lagrange’s set of 
conditions : * 
R = 0, 
02R _ 0 *-^ 
da 2 m ’ ’ ba k ~ l 
0. 
Trudi, N. (1862). 
[Teoria de’ Determinant:, .... xii + 268 pp. Napoli.] 
To Trudi is due the first methodical exposition of bigradients, a nineteen- 
page chapter of the first part ( Teoria ) of his text-book being specially 
devoted to them, and several chapters of his second part (Applicazioni) 
making constant use of them. 
The nineteen-page chapter or section (§ xi., pp. 94-112) bears the head- 
ing “ Matrici e determinanti a due scale.” It contains, first of all, careful 
explanations of the various expressions which he finds necessary to use in 
* Lagrange. Reflexions sur la resolution algebrique des equations. Nouv. Memoires 
. . . Acad. . . . Berlin , 1770, 1771 : or (Euvres completes , iii. pp. 205-421 (227-229). 
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