483 
Proceedings of Poyal Society of Edinburgh. [july 18 , 
en resulte que le produit d’un nombre quelconque de fonctions, 
telles que %{y, z')(v, £'), est lui-meme de la forme (y, z ) .” 
The application here of the identity 
%ab' — %a%b - % ab 
requires a little attention. The result of multiplication and classi- 
fication of the terms is 
%yv.z'Z - tzv.yf , 
or, as it might preferably be written, 
%{yv .z£) - {tzv .y(}) 
and this we know from the said identity 
= \%yv . 24 - %(yv . z£j] - [ 22 V . 2 \yl - 2(zv . ?/£)] , 
which, because of the equality of 2 (yv . 4) and 2 (zv . yt), becomes 
2 yv . 2z£ - 2 zv . 2 yl . 
The inherent weak points, however, of the mode of demonstration 
stand out more clearly when the next case comes to be considered, 
viz., the case for resultants of the third order. From the three sets 
of n letters 
t ft 
X 9 X, X 9 ..... 
?/> y\ v", 
f n 
2 , 2 , 0 , ..... 
all possible “ r4sultantes a trois lettres ” are formed, and each re- 
sultant is multiplied by the corresponding resultant formed from 
other three sets of n letters , 
t, c, r, 
r n 
V, V, V , 
z, t, r, 
Each of these \n(n - l)(w— 2) products consists of 36 terms, there 
being thus 6n(n - 1 )(n - 2) terms in all. But these Qn(n - 1 )(n - 2) 
terms are found to be separable into six groups, viz. 
+ 2 {xS.y'v'.JT}, + 2 {^. 2 V.^T}, .... 
so that the result which we are able to register at this point is 
2 {xgj X') = 2*£ • yv . z"£ " + 2 y ( . 2 V . *"£" 
+ 2 zg.afv'.y'T -2 xPdv' .y'X' 
-2 yi.dv’ .W-M.tfv' 
