1887 .] Dr T. Muir on the Theory of Determinants. 467 
equations, there is not so much that is noteworthy. Methods 
previously known are followed, the new features being formality 
and rigour of demonstration. 
The coefficients of the equations being 
11, 12, 13, .... , lr 
21, 22, 23, .... , 2r 
rl, r 2, r3, . . . . , rr 
it is noted, as Vandermonde had remarked, that the common 
denominator of the values of the unknown may be got in two ways, 
viz., by permuting either all the second integers of the couples, 
11, 22, 33, . . . . , rr, or all the first integers : but this is supple- 
mented by a proof, that if any term be taken, e.g., 
16.24.33.47.51.68.79.82.95 
with the couples so arranged that the first integers are in ascending 
order, and the sign be determined from the number of inversions in 
the series of second integers, then the sign obtained will be the same 
as would be got by arranging the couples so as to have the second 
integers in ascending order, and determining the sign from the 
inversions in the series of first integers. The proof rests entirely on 
the previous theorem, that conjugate permutations have the same 
sign ; indeed the new proposition is little else than another form of 
this theorem. (in. 14.) 
The desirability of an appropriate notation for the cofactor, which 
any one of the coefficients has in the common denominator is 
recognised,* and the want supplied by prefixing f to the coefficient 
in question ; for example, the cofactor of 32 is denoted by 
f 32. 
It is thus at once seen that the denominator itself is equal to 
In. fl n + '2n.{2n +....+ rnSrn, 
or n \ . TM + ?z2.fw2 +. . . .+ nrSnr. (vi. 2.) 
Also by this means one of Bezout’s (or Vandermonde’s) general 
theorems becomes easily expressible in symbols, viz., 
Iw.flm + 2n.f2m +.... + rn.frm = 0, (xil 4.) 
* Lagrange’s use of a corresponding letter from a different alphabet must 
not be forgotten. 
