1887.] Dr T. Muir on the Theory of Determinants. 
489 
To the right hand member of this the substitution 
%ab r c" = + 2 %dbc — 'ZaSbc - %b%ca - %c%ab 
is now applied six times in succession ; that is to say, for 
^ J- If If <>!! 
Axt , . y v . z 4 
and the five other term-aggregates which follow, we substitute 
+ 2%(oc £ . yv . z£) 
- . z£) - %yv$(zt . x£) - • yv) 
and five other like expressions. By this means we arrive, “ toute 
reduction faite,” at 
%{x,y\z")(&'£') = %x&yvM + ty&zvtx^ + %s&xv$yZ 
- %xgfav%y£ - %yi%xv%z'Q - %z£$yv%x£ > , 
which is the result desired. 
It is easy to imagine the troubles in store for any one who might 
have the hardihood to attempt to establish the next case in the 
same manner. 
If Binet’s multiplication-theorem he described as expressing a 
sum of products of resultants as a single resultant , his next theorem 
may he said to give a sum of products of sums of resultants as a 
sum of resultants. The paragraph in regard to it is a little too much 
condensed to be perfectly clear, and must therefore he given 
verbatim. It is (p. 288) — - 
“ Designons par S (y'f) une somme de resultantes, telle que 
(y/.0 + (y„W) + (y + &c - ; 
e’est-a-dire, 
y, z , ~ z ,v, +y„ z „ ~ z „y„ + y,„ z ,„ 
z „',y„" + ; 
et continuous d’employer la caracteristique S pour les integrales 
relatives aux accens superieurs des lettres. L’expression 
^[S(y,^) . S(v,£')] devient par le developpement de chacun de 
ses termes, et en vertu de la premiere formule de l’art. 1 ou de 
celle du no. 4 , 
+%„ »>„£, - +&c. 
+ 2z,c„ - 3z,v„5 y,i„ + %y - 2z„u„%X, + &o. 
+ &C. 
