568 
ME. A. CAYLEY ON TSCHIENHAIJSEN’S TEANSEOEMATION. 
The determinant in question was calculated by the formula 
Div. 
□ =-12.345 1 
+13.245 2 
-14.235 3 
+15.234 4 
-23.145 5 
+24.135 6 
-25.134 7 
-34.125 8 
+35.124 9 
-45.123 10, 
where the duadic symbols refer to the first and fifth columns, viz. 12 is the determinant 
formed out of the lines 1 and 2 of these columns, and so for the other like symbols ; 
and the triadic symbols refer to the second, third, and fourth columns, viz. 345 is the 
determinant formed out of the lines 3, 4, 5 of these columns, 'and so for the other like 
symbols. 
The ten divisions were separately calculated. It is to be noticed that these divisions 
other than 4 and 6 correspond to each other in pairs, while each of the divisions 4 and 
6 corresponds to itself, as thus : 
Div. 1, -10 
2, - 9 
3, -7 
5, - 8 
4, - 4 
6 , - 6 , 
viz. if in the place of 
y ) f i 
we write 
—y ; /, e, d, c, b, a; E, D, C, B, 
then division 1 becomes division 10 with its sign reversed, and so for divisions 2 and 9, 
3 and 7, 5 and 8 ; while each of the divisions 4 and 6 is unaltered, except that the sign 
is reversed. But the corresponding divisions were each of them calculated, and the 
property in question was used as a verification. Another very convenient verification, 
which was employed for the several divisions, was obtained by putting 
«=5=c=6Z=e=/=B=C=D=E=l, 
upon which supposition the determinant becomes 
