149 
1906-7.] Dr Muir on Axisymmetric Determinants. 
unnecessary, it has now to be noted that Sylvester had the determinant- 
form of W before him throughout: he even says that he had tried to 
express the quotient as a determinant, but had been unsuccessful. Without 
further restriction, then, as to form, his proposition is If F , G , H , K be the 
complementary minors of the elements in the places 11 , 22 , 33 , 44 of the 
determinant 
(ab) 2 ( ac ) 2 (ad) 2 1 
(ab) 2 • (be) 2 (bd) 2 1 
(ac) 2 (be) 2 • (cd) 2 1 
(ad) 2 (bd) 2 (cd) 2 ■ 1 
1111 . 
or 2W say, 
then the result of rationalising 
n/F + :JG + x/H + Jk 
is divisible by W. This is not all, however ; for Sylvester having noted 
the analogous case connected with the relation between the perimeter and 
area of a triangle, namely, the fact that 
(ab) 2 (ac) 2 1 
(ab) 2 • (be) 2 1 
(ac) 2 (be) 2 • 1 
111 - 
i.e. 
is a divisor of the result of rationalising 
J2 Jbcj 2 +J2 (mf +JIfab) 2 
where the radicands are the complementary minors of the elements in the 
places 11 , 22 , 33 of the determinant, boldly extends the proposition (with- 
out proof) to any triangular number of arbitrary quantities, taking 
occasion also to point out that when we leave geometry (ab ) , (ac ) , . . . . 
may be written for (ab) 2 , (ac) 2 , . 
Hesse (1853, April). 
[Ueber Determinanten und ihre Anwendung in der Geometrie, 
insbesondere auf Curven vierter Ordnung. Grelle’s Journal , 
xlix. pp. 243-264.] 
The main subject of the first half of Hesse’s paper (pp. 243-253) is a 
property of axisymmetric determinants required for the establishment of 
the geometrical results contained in the second half. In the first three 
pages he considers the relations between the minors of two general 
