Dr T. Muir on the Theory of Determinants. 
513 
(B"") 
(C"") 
r\/Jc 1 .. . 
mp\ f p r^ 
c dj ~ ( a c? 
)(a h 
. .mp\ ( p T\( 
■ . c/j + [ae)( 
' k 1 . . .mp\ 
ah . . . df ) 
. . . . + 
fp r\ f kl . 
[ah j ya . . 
. . m p 
.cdJ 
mp r\ 
cdj) 
( 
7c p r\f k l . . , 
a h fj\a h . . . 
. mp\ 
:cd) 
/ 7c p r\7 k 1 . . 
~ yah d jya h . . 
.mp\ 
■Cf) 
/ Ic p r\/ 7c 
'^yahc jya 
l . . . mp\ 
b...df) 
/7c p\fk l 
. . .mpr\ 
...cdf)- 
and the general result including them is referred to. (xlvi.) 
That they are extensionals of the definition is evident from the fact 
that the index may be moved to the left so as to make ^ common 
to every factor of (B""), and ^ common to every factor of (C"") . 
Still another series of results is obtained, but they are essentially 
the same as the foregoing, the difference again being merely a 
matter of rows and columns. 
All these preparations having been made, Desnanot returns to 
the subject of the relations between the numerators and denominators 
of the values of the unknowns in a set of linear equations. Thirteen 
pages are occupied with the case of four unknowns, the number of 
relations found being 74, of which, after scrutiny, 14 are retained. 
The case of five unknowns, and the case of six unknowns are gone 
into with about as much detail, and then, lastly, the general set 
of 71 equations with n unknowns is dealt with. None of the 
relations obtained need be given, as they are all included in the 
identities which have been spoken of above as extensionals. 
The second chapter (p. 94) bears the heading 
Simidification des formides generates qui donnent les 
valeurs des inconnues dans les equations du premier 
degre, lorsqiion vent les evaleur en nomhres. 
Here again the cases of three, four, five, six unknowns are dwelt 
upon with equal fulness in succession. The consideration of one of 
them will suffice to show the nature of the method, and will enable 
the reader to judge of the amount of labour saved by employing it. 
Choosing the case of four unknowns, we find at the outset the equa- 
tions stated and the solution condensed as follows (p. 104) : — 
19 / 2 / 89 ] 2 L 
VOL. XV. 
