Dr T. Muir on the Theory of Determinants. 
515 
having been calculated, the identity 
\ah'\ . D = \ah'c"\ . \ah'd"'\ - \ah'c"'\ . \ah'd"\ 
is used to find D. 
(4) A similar identity 
\ah'\ . N'" = \ah'cf . \ah'f'\ - \ah'c"'\ . \ah'f"\ 
is used to find N'". 
(5) A similar identity 
ja&'I.N" = \ahf"\.\ah'd'"\ - \ah'f"\.\ah'd"\ 
is used to find N". 
(6) Two subsidiary quantities S, S' are calculated, the first being 
= f\ctb'c"d"\ - c\alj'fhr\ - d\ah'c"f"'\, 
and the second 
= /' \aVc"d"'\ - c' \ah'f"d"'\ - d'\cib'e"f"\ . 
(7) From these W and hi are readily got. For evidently 
aS' - Sa' 
= \af'\ . \ab'c"d"'\ - |ac'| . \ah'fhr\ - \ad'\ . \ab'cj”'\ 
and this by a previous theorem 
, = \ab'\ . \afc"d '"\ , 
= K1.N'. (XIIL 3.) 
The third chapter consists of a lengthy examination (pp. 157-264) 
of the singular cases met with in the solution of linear equations, 
and does not at present concern us. 
CAUCHY (1821). 
[Cours d’ Analyse de I’Ecole Royale Polytechnique I. xvi. + 576 pp. 
Paris.] 
When Cauchy came to write his Course of Analysis, afterwards 
so well known, he did not fail to assign a position in it to 
the subject of his memoir of 1812. The third chapter bears the 
heading, Des Fonctions Symetriques et des Fonctions Alternees.'’^ 
