Dr T. Muir on the Theory of Determinants. 507 
Having thus shown that if in each of the second factors of the 
identity 
a new letter c be added and the index 1 be prefixed, the sign of 
equality may still be retained, so that we have a new identity 
- \af^\af.f^\ + \af^\af.^c^\ = 0 (B); 
he then goes on to prove in the same fashion that the first factors 
of this derived identity may be treated in a similar way with 
impunity, viz., that they may be extended by the appending of the 
letter c with a new index 5, so that we have a further derived 
identity 
1*^1 ^ 2 ^ 5 ! *^ 3 ^ 4 ! “ !^1^3^5ll‘^h^2^4! d- l<^i^4C5|kq^2^3l “ ^ (^)’ 
already known to us from Monge. 
And this is not all, for the next paragraph shows that these two 
extensions may be repeated in order as often as we please, the 
opening of the paragraph being as follows (p. 15) : — 
“ 17. Generalisons et prouvons que si la forinule 
/ k l . . q\h km . . 'p\ h km. . l\h k p . . q\ _ fk I . . p\/ km . . q\ 
\ah . . c J\ct h . . cj ~ \cib . . c j\ci h . . cj ~ \ajh . . c J\a h . . cj 
est vraie dans le cas on il y aurait n lettres comprises dans 
chaque facteur, elle sera encore vraie en ajoutant une nouvelle 
lettre d dans les seconds facteurs de chaque terme avec I’indice 
I qui n’y entre pas ; et qu’ensuite, si Ton ajoute la meme 
lettre d dans les premiers facteurs de chaque terme avec un 
nouvel indice ?■, TegaHtd ne sera pas troublee. 
II s’agit done de demontrer que ces deux formules sont 
exactes : 
/ kl . . q\f k m .. I p\ _ /km . . l\/ k p . . I q\ _ / k I . . p\/ k m . . I q\ 
yah .. c h . . cdj \cl h . . c j\a b . . cdj ~ \ab . . c Jya b . .ed) 
(xxiii. 4 ) 
( k l . . q r\/ km .. I p\ / km .. I A / k p . . I q\ _^ / k I . . p r\ / km.. I q\ 
ab . . ed j\ct b . . edj ~ \ab . . ed J\a b . . edj ~ yab . . ed J\a b . . e d) ^ 
The line of proof is still the same, and may be shortly indicated by 
treating the case 
(D) ~ 1^1^2^4ll*^1^^2^3^5l d" l^i^2^3ll^l^'^2*^4^^5l “ ^ J 
