496 
Proceedings of Boyal Society of Edinburgh. [july 18 , 
^\Wh\ > I x \y?> I }" ~ ^ I P • ^ I x \V% I 2 (^i^i^l I ^1^/2 1 ) 2 ’ 
where the elements of the resultant on the right are sums of 
products of quantities in the second triad of rows. Then the same 
theorem is used to make a further step backwards, viz., to express 
each of these three sums of products of resultants as a resultant 
whose elements are sums of products of the quantities in the first 
triad of rows, the effect of the substitution being 
2||zi* 2 I > I ! | 2 = — (Sz 1 * 1 ) 2 }{S* 1 2 S2/ 1 2 — (Say/i) 2 } 
- \$z 1 x l %x ] y l - jS*! 2 } 2 . 
Simple multiplication transforms this into 
2x 2 1 S^S^Szj 2 - ty^{%z ft) 2 - ) 
' 1 ( + 2’Sflj l z 1 'S,z 1 x{%x l y 1 - 2x 1 2 (Sj/ 1 z 1 ) 2 > ’ 
which, by still another use of the multiplication-theorem, we know 
is equal to 
%\x x yfa 'j 2 . 
The set of six results of which this is one, is 
SXj 2 =S* 1 2 S|a: 1 ?/ 2 z 3 | 2 , 
SYj 2 = S^ 2 2| Ws | 2 , 
tZ? =Sz 1 2 S|cc 1 2/ 2 z 3 | 2 , 
SYjZj = S^ZjSta^Zgl 2 , 
SZjXj = 2z r « 1 S!x 1 :y 2 z 3 ! 2 » 
SXjY 1 = 
if, for shortness, we denote the quantities of the third triad of 
rows by 
-^2> 
Y 1( Y 2 , 
z 2 , 
Following these, and deduced by means of them, is an equally 
noteworthy theorem regarding the sums of squares of all the result- 
ants of the third order, which can he formed from the quantities 
of the second triad of rows, Denoting these quantities tempo- 
rarily by 
£i> & 
Vi> 
£u £2’ 
(xxxv.) 
