1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
421 
doubt, however, that it was the second fact alone, — in other words, 
the discoveries of Jacobi, Sylvester, and Bichelot, — which influenced 
the veteran Cauchy to return to a subject practically untouched by 
him for thirty years. 
The opening part of the paper is, of course, necessarily old matter. 
One thing to be noted is that Cauchy tacitly discards the term 
determinant , which he was the means of introducing, using uniformly 
the more general expression fondion alternee instead. Another is 
that he adopts the rules of signs which makes use of the number of 
interchanges. From this his own peculiar rule of signs is deduced, 
and made the starting point for the fresh investigation which forms 
the main portion of the paper. The exposition of his rule, which 
differs from that of 1812, is worthy of a little attention, both on its 
own account and because otherwise the matter following would be 
scarcely intelligible. In the case of any term (“ terme ” or “ pro- 
duit ”) of the determinant 
say the term 
^ — a 0,Q a l,i a 2,2 a 3>3 a 4cA a 5,5 a 6,G 5 
^0.l a L0 a 2.5^3.3 a 4.6 a 5,4%.2 ’ 
there is an underlying separation of the indices 0, 1, . . , 6 into 
groups (“ groupes ”), by reason of the system of pairing; that is to 
say, since an index is found paired along with one index and not 
with another, there arises the possibility of looking upon those 
which happen to be paired with one another as belonging to the 
same family group. Thus, attending to the first a of the term, we 
see that 1 and 0 belong to the same group, and as on scanning the 
rest of the term, we find neither of them associated with any other 
index, we conclude that the group is binary (“ un groupe binaire ”). 
Again, we see that 2 is paired with 5, 5 with 4, 4 with 6, and 6 
with 2 ; this gives us the quaternary group (2, 5, 4, 6). Lastly, 
3 is seen to be paired with 3, and thus forms a group by itself. 
Now, if we wish to find how many interchanges of the second 
indices are necessary in order to obtain the given term 
^O.l^l.O a 2,b a 3,3 a ±,Q a bA a G,2 
from the typical term 
^O.O^l.l a 2,2 a 3,3 a 4A a 5,b a 6,G > 
we may do the counting piecemeal, attending at one time to only 
