396 
Proceedings of the Royal Society of Edinburgh. [Sess. 
XXIY. — The Theory of Bigradients in the Historical Order of 
Development np to 1860. By Thomas Muir, LL.D. 
(MS. received January 22, 1910. Read February 7, 1910.) 
As we have already pointed out (Hist., i. p. 487), bigradients were first 
brought to light by Sylvester in 1840 in the paper in which he made 
known his so-called “ dialytic ” method of eliminating the unknown from 
two equations of the same or different degrees. Shortly afterwards 
Richelot and Cauchy recalled attention to Euler’s and Bezout’s method of 
1764, as giving substantially the same result as Sylvester’s, the fact being 
that the determinant obtained by Sylvester differs from that obtainable in 
the other case merely by being its conjugate. The details of these papers 
and of others related to them have already been given. 
Cayley, A. (1844). 
[Note sur deux formules donnees par MM. Eisenstein et Hesse. Crelles 
Journ., xxix. pp. 54-57 ; or Collected Math. Papers, i. pp. 113-116.] 
Although Eisenstein’s property * of the discriminant 
a 2 d 2 - 3b 2 c 2 + 4ac 3 + 4 bhl - babcd , or A say, 
of the binary cubic ax z -f 3bx 2 y 4- Sexy 2 + dy 3 , — namely, the property that 
A 2 D 2 - 3B 2 C 2 + 4AC 3 + 4B 3 D - 6ABCD - (a 2 d 2 - 3b 2 c 2 + Aac* + ±bhl - 6abcd)* 
when 
i, B, C, D 
= - 
X 0A 
2 a d 
j0A 
’ 6 fo’ 
n 0A 
*db 
X 0A 
’ 2 0a ’ 
sed in the form 
A 2B C 
a 
2 b 
c 
A 
2B 
c 
a 
2 b c 
B 2C 
D 
b 
2 c 
d 
B 
2C 
D 
b 
2c d 
it is not as a relation between two four-line determinants that it has 
been studied. 
Cayley in effect says that if we wish to find substitutes A, B, C, . . . for 
a, b, c, . . . so that 
<M A, B, C, . . .) = {^(a, b, c, . . .)}* , 
* Crelle’s Journ., xxvii. pp. 105-106, 319-321. 
