Note on Hesse s Generalization of Pascal's Theorem. 
41 
By adding the first four rows one by one in order to the last four rows, 
and then subtracting the 6th, 7th, 8th, 9th columns from the first four 
columns in similar fashion, this is readily transformed into 
lU^ ID. e. 
which is seen to break up into 
Vj_ V2 v.. 
4i 
42 
^3 
/^3 
^4 
C2 
C3 
?<3 
u. 
a I 
7i 
a 2 
72 
£3 lU^ 
?t?3 
"3 
73 
X2 
X, 
«4 
74 
^3 
■ /3i 
/^3 
/^4 
C2 
C3 
C4 
^'4 
«i 
7i 
^'2 
^3 
^'4 
a 2 
72 
li\ 
ILK 
a. 
73 
£4 
^3 
^4 
«4 
74 
Further, the seven-line determinant here, by having its first column made 
the last and its 1st, 2nd, 3rd rows the 7th, 5th, 6th respectively is seen to 
be the bordered determinant which Hesse would have denoted by 
[aye,lDc'L,\ . 
Our final result thus is 
[£,;] . [ay,;3h] + [£,/3] . [ay,04] + [e,c] [ay,^ 
\u,,,\.[aye,(3ci;] (I) 
4. By transposition of the column of e's with the column of a's and 
the changing of signs throughout, (I) takes the form 
l2tiJ.[ay£,/3?4] = [a,/3J . [y£,c4] + [a,c] . [y£,/3;] + [a,4] . [y£,/3c] , 
and is then seen to be the second of a series of identities of which Hesse 
gave the first, namely, 
\n,,\.[ay,l'DC'] = [a,/3].[y,c] - [a,c^] . [y,/3] . 
