403 
1909-10.] Dr Muir on the Theory of Bigradients. 
abed 
abed. 
)> q r 
• p q r 
p q r . . . 
The two-line minors of the first two rows of this determinant being denoted 
by 12, 13, . . . , 45, and the three-line minors of the remaining rows by 
123, 124, . . . , 345, it is seen that 
R 3 ,2 = 12-345 - 13-245 + 14*235 - 15*234 
+ 23*145 -24-135 + 25’134 
+ 34-125 - 35-124 
and it will be found that 
-45123 
12-345, - 13-245, 14-235 + 23-145, -(15-234 + 24-135), . . . 
correspond to the 1 st , 2 nd , 3 rd , 4 th , . . . squares of Cayley’s expression. 
The paper closes with the six resultants R 2)2 , R^, R 4 , 2 , R 3 , 3 , R 4)3 , R 44 
printed each in the new form as a chain of squares;* they occupy four 
quarto pages. 
Zeipel, Y. v. (1858, June). 
[Demonstration of a theorem of Mr Cayley’s in relation to Sturm’s 
functions. Quart. Journ. of Math., iii. pp. 108-117, or Nouv. Annates 
de Math., xix. pp. 220-224.] 
The theorem referred to is that of August 1848. Zeipel’s proof (pp. 1 OS- 
114) is lengthy and unattractive, and scarcely warrants reproduction. The 
remaining pages are occupied with the curious identities 
Pr- 1 
P r 
Pr-1 
Pr 
P'r-1 
P'r 
Pr 
= -PVlPr+1, 
P'r-1 
P'r 
Pr 
PVl 
P"r 
P'r 
P "r- 1 
P'r 
P"r 
and the corresponding identities in which the left-hand members are of a 
higher odd order than the third. 
Hesse, O. (1858, October). 
[ii determinante di Sylvester ed il risultante di Eulero. Annali di Mat. . . . 
ii. pp. 5-8; or Werke, 475-480.] 
The determinant referred to is that to which Sylvester was led in 1840 
by his so-called “ dialytic ” method, and which, as we have already seen, 
* In the expression for R 4>4 there is at least one misprint, namely, a 2 c 2 for a 2 b 2 outside 
the third square of the chain. 
