162 Proceedings of the Royal Society of Edinburgh. [Sess. 
as a consequence of the relation 
2 2 (*! - x z)( X \ - x s) = 2 (*1 - *2) 2 + 2 K - ^s) 2 - 2 (*« “ ■ 
Another form of the relation between the mutual distances of four points 
in a plane is thus brought to light. 
Rubini (1857). 
[Applicazioni della teorica dei determinanti. Annali di sci. mat. efis., 
viii. pp. 179-200.] 
Rubini starts with the theorem which expresses a determinant with 
binomial elements as a sum of determinants with monomial elements, and 
then considers a long series of special cases. Among these Ferrers’ 
theorems of the year 1855 occupy the first place (§§ 2, 3, pp. 181-184). 
Bellayitis (1857, June). 
[Sposizione elementare della teorica dei determinanti. Mem. . . . 
istituto veneto .... vii. pp. 67-144.] 
Besides the paragraphs (§§ 42, 43, 44) specifically devoted in the second 
half of the exposition to axisym metric determinants, there are two others 
(§§ 9, 35) connected with the same subject in the first half. One of the 
latter (§ 35) draws attention to the fact that any coaxial minor of the 
axisymmetric determinant which is the square of a determinant is 
expressible as a sum of squares. What is new in the former is the definite 
reference to determinants which are doubly axisymmetric (“ doppiamente 
simmetrici ”), the examples given being * 
a 
b \ 
a 
b 
c 
a 
b 
c 
d 
b 
<*i. 
b 
d 
b 
b 
a 
d 
c 
c 
b 
a 
5 
c 
d 
a 
b 
d 
e 
b 
a 
the last of which is noted as being equal to 
- (a + b + c - d)(a + b - c + d)(a - b + c + d){ - a + b + c + d). 
* The third, by reason of its central two-line minor which might have been 
more specialised than a doubly axisymmetric determinant. 
