538 
Proceedings of Royal Society of Edinburgh. [sess. 
Pi P 2 Pi i 
7i 72 7s I » 
C 1 C 2 C i 
C 1 C 2 C 3 
C 1 C 2 H 
Pi P2 Pi 
7l 72 73 
\ \ h 
d-^ d^ dg 
d-^ c ?2 dg 
6^2 6^2 ^3 
ft ft ft 
7i 72 7s 
S 2 S 3 
They might even be used as such alongside of quantitative 
elements in the same determinant ; and there is at least one case 
of this where the advantage is most striking : indeed it is not too 
much to say that we are thus enabled to express with fulness and 
accuracy a famous theorem of Binet’s which up till now has 
remained unformulated. Binet in 1812 ( Journ . de V Ec. polyt., ix., 
cah. 16, p. 284) says, “ On verifie aisement les formules suivantes 
'Zab' = ^al,b - %ab, 
%ab'c” = ^aZb^c + 2 ^abc - ^ a%bc - ^b%ac - 'Sc'Sab, 
'Zab'c'd'" = 'Xd%b%c%d - ttabcd 
- %d%b^cd - ^a%c%bd - %d%d%bc 
- %b%c%ad - 'Ib'^d’^ac - '%c%d%ab 
+ ^ab^cd + 2 ac%bd + ’Sadlbc 
+ ’2%a^bcd + 2 '^b'Xcda + 2 ^c%dab + 2 %d%abe, 
%ab'c"d'"e!"' = %a%b%cZd%e + .... 
\ 
h 
\ 
^2 
&3 
ft 
ft 
CO 
oa. 
7i 
72 
73 
O 
O' 1 
to 
h i 
% 
C 2 
c i 
c i 
C 2 
c i 1 
C 1 C 2 
ft 
ft 
ft 
7i 
72 
7s 1 
\ 
\h 
h 
h 
6j b. 
2 ^3 
'CO 
to 
Pi 
7i 
72 
73 
1 
*1 & 
2 ^3 1 
The law of formation of the right-hand members was left un- 
divulged: and probably Bellavitis in 1857 was the first to draw 
attention to the fact that the said members bear a wonderful 
resemblance to the final expansions of axisymmetric determinants 
{Sposizione element are . . . . §91); but he only got so -far as to 
say that in order to complete Binet’s fourth instance “lo sviluppo 
del determinante simmetrico ” of the fifth order must first be 
found, and then certain arbitrary changes made therein. With 
