1888 - 89 .] Dr T. Muir on the Theory of Determinants . 
425 
are derivable from different permutations of the seven indices 0, 1, 
1, 3, 4, 5, 6. In fact, the / groups of one index each may be per- 
muted among themselves in every possible way, so may the g binary 
groups, the h ternary groups, &c. Further, with like immunity to 
the term, each separate group may be written in as many ways as 
there are indices in it, — the group (4, 5, 6), for example, being 
safely changeable into (5, 6, 4) or (6, 4, 5). The number, there- 
fore, of different permutations of 0, 1, 2, 3, 4, 5, 6, which will give 
rise to any particular term, is 
(1.2.3.../x 1.2.3. m .gx 1.2.3 .. .hx ... x 1.2.3...?) x (1/2*3*... w*) , 
or say, 
(f\g\h\. . . ?!)(1W. ..»*). 
There thus results the equation 
(f\g\h\. . ./!)( 1W. . • n l ) N/, , , * = h\, 
whence 
!= (f\g\ hY. . . Z!)(l'2»3*. . . ' ^ LV ' ^ 
Following this interesting result a few deductions and verifica- 
tions are given. First of all it is pointed out that since the total 
number of terms of all species is n ! we must conclude that 

where f+2g + 3h + ...+nl = n. 
Cauchy says (p. 423) : — 
“ Cette derniere formule parait digne d’etre remarqueb. Si, 
pour fixer les idees, on prend n = 5 l’equation donnera 
1 . 2 . 3 . 4 . 5 = N 5 i 0 , 0,0.0 + ^"3,3 , 0 , 0,0 + N"l,2, 0,0,0 + N" 2>0( 1,0,0 
4 " -N"o, 1,1, 0,0 + Nl, 0,0, 1,0 + No, 0,0, 0,1 5 
et par suite 
1 . 2 . 3 . 4 . 5 = 1 + 10 + 15 + 20 + 20 + 30 + 24 = 120 , 
ce qui est exact.” 
Again, since the number of positive terms in a determinant is 
equal to the number of negative terms, and since the terms, whose 
number ...n has .just been found, have all the sign-factor 
