228 
Proceedings of Boyal Society of Edinburgh. [sess. 
ants of the 2nd order, and (to save on the breadth of the page) 
denoting 
in -j Ctf b,C, .. . 
aa + b^ + cy + . . . by ’ 
a,(3,y , . . . 
we should have 
I a A C 3 I • I a l/^273 I 
= «l a l 
^2’ C 2 
&’72 
^2? C 2 
@3>73 
a 2 ttj 
^l» c l 
P2^2 
V?i 
Aj>78 
+ agCtj 
^l> C 1 
P&72 
c i 
@3’73 
^3> C 3 
^3> C 3 
^3> C 3 
^3» C 3 
^2» g 2 
^2» C 2 
$2?l2 
ft>73 
A’ 72 
Ap7b 
@2)72 
^3>73 
- a 1 a 2 
^2> C 2 
ft>7i 
^2> C 2 
/^3’73 
C? 2 0t 2 
& 1> C 1 
ft>7i 
^1> C 1 
/^3>73 
— %a 2 
hlfl 
Pv7i 
/?3>73 
^3’ C 3 
^3j 
^3> C 3 
^3> g 3 
bjpf 2 
^2> C 2 
&>7i 
ft’73 
/^l»7l 
ft’73 
£l>7i 
/^3>73 
+ ^l a 3 
b<z> c 2 
ft»7i 
5 2 , c 2 
ft’72 
— $ 2 a 3 
A»7i 
$2?l2 
+ %a 3 
&i, Cj 
Al>7i 
^1> C 1 
/^2’72 
bsi c 3 
^3> ^3 
^3> C 3 
^3> C 3 
^2> g 2 
^2> g 2 
ft>7i 
ft’72 
ftiTi 
A’72 
Pv7i 
P2’72 
That each line of this result is not altered in substance by writing 
c 2 £ or 
a 2’/^2>72 
^2> C 2 
P&72 9 
%>^2> C 2 
a 3’/^3’73 
for &c„ 
P3>73 
would probably be shown by expressing the line in the form of a 
determinant of the 3rd order, e.g ., the first line in the form 
b v Cj 
&1, Cj 
$2*72 
03>73 
^2 > g 2 
^2? g 2 
02*72 
Ps’7s 
^3> g 3 
^3’ C 3 
02>72 
03>73 
and increasing each element of the second column by a 2 times the 
corresponding element of the first, and each element of the third 
column by a 3 times the corresponding element of the first. The 
whole result would in this way be transformed into 
