494 
Proceedings of Boyal Society of Edinburgh. [july 18, 
2 | I 2 
= $y 2 2\h x 2 1 2 + Sz 2 S|*i 2/ 2 I 2 - Sa^ly^l 2 + ^y^\h x 2\\ x iV^ , ) 
= %z>% \xggff + l i x 2 %\y l z 2 \ 2 - %y 2 %\z x x 2 ^ + 2^2^|x ] ?/ 2 |j^ 2 | , Wxxxm.) 
= | 2 + Sy 2 ^^ | 2 - %z 2 %\x l yf‘ + 23 ^ 31^2 |bf£ 2 l , J 
and immediately from these the set 
2 K 2/ 2 %| 2 =^ 2 %^ 2 ! 2 + ^^fiVzfydf + 2 ^% iZ 2 M ^ 2 l>i 
= ^ 2 E|^ 2 | 2 +. txytly^fz^ + tyz^z^.fgg^ , l (xxxiv.) 
= %z 2 %\x 1 y 2 \ 2 + %yz$\zjx 2 \.\x$ 2 \ + 'Zzx%\x 1 y 2 \.\y 1 z 2 \ . ) 
We may note in passing that either of these sets leads at once to 
the initial theorem 
$%\x 1 y#$ = '2fr 2 %\y 1 z 2 f + % 2 2 1%^ 2 | 2 + 2z 2 2Ky 2 | 2 
+ 2%yz%\z 1 x 2 \.\x 1 y 2 \ + 2%zx%\x 1 y 2 \.\y 1 z 2 \ 
+ 2%xy%\y 1 z 2 \.\z 1 x 2 \ , 
and that with the multiplication-theorem already established this 
reverse order would be the more natural. 
The next step taken is the formation of resultants of the 2 nd order 
from elements which are themselves resultants of the 2 nd order ; 
viz., just as from the three rows of n quantities 
ry» ry* ry> ry* 
*^1 *^2 ^3 • 9 9 • 
Vl V2 ds • • • • Vn 
ci ^2 ^3 • • • • ^ n 
there were formed the three other rows of fj n(n- 1) quantities 
\Vl Z 2\ > 1 ^ 1 % 1 » ' 
• • • 5 \vM ? (^ 2 ^ 3 ) ? • • * 
i ^\ X 2 1 ) !%^ S I ’ * 
.... 1 z,x m \ , \z-xd\ . . . . 
, . . . 2 t X A 
5 1 1 W 1 5 | 2 3 15 
K?/ 2 l 5 | *^l 2/3 i J * 
• • • J 1 X \V n | ) |*^ 2^3 1 ’ * * 
• • • , K-l^nl 
so from the latter three other rows of quantities 
bi a? 2 | 
I^i^sl 
| - 2^n| 
| ^"n - lAil 
l^y 2 l 
l^l^i 
K- 2 &J 
!''-n - l^n | J 
K2/2I 
fci-SS&J 
K-lJ/nl 
te! 
\Vih\ 
|2/«-2^w| 
li/n-l^w) } 
|yAl 
|2/i z s! 
l2/n-2^nl 
1 Vn 1^«| 
1^1 ^2! 
K ^3 1 
k- 2^1 
bn - 1 ^nl 5 
