1888 - 89 .] Dr T. Muir on the Theory of Determinants. 
423 
Also, the full expansion of a determinant may he represented hy 
writing a term of each cyclical species , and prefixing to each such 
typical term the symbol 2 icith its proper sign, + or — . (lv. 2) 
To obtain a term of any given cyclical species, that is to say, 
corresponding to given values of f g, h, . . . , all the preparation 
that is necessary is to write the indices 
0, 1, 2, 3, \ , (7.-1), 
enclose each of the first / of them in brackets, enclose in brackets 
each of the next g pairs, then each of the next h triads, and 
so on. This gives the groups of the term, and the term itself readily 
follows. For example, if we desire in the qase of the determinant 
2 ± ^oo a u tt 22 a 33 a 44 a 55 a 66 a term corresponding to /= 2, g = 1 , h= 1* 
we take the indices 
0, 1, 2, 3, 4, 5, 6 ; 
bracket them thus 
(0), (1), (2, 3), (4, 5, 6); 
and with the help of this, write finally 
a 0,0 a h\ a 2,3 a 3.2 a 4.5 %4 ' ( IL *0 
The number of different cyclical species of terms in a determin- 
ant of the 77 th order is evidently equal to the number of positive 
integral solutions of the equation 
/+ 2^+3/.+ . . . + nl — n . (lv. 3) 
Cauchy’s illustration of this is clearness itself. He says (p. 419): — 
“ Si, pour fixer les id4es, on suppose n = 5, alors, la valeur 
de n pouvant etre presentee sous 1’une quelconque des formes, 
1 + 1 + 1 + 1 + 1 , 
1 + 1 + 1 + 2 , 
1+2 + 2 , 
1 + 1 + 3, 
2 + 3, 
1 + 4, 
5 ’ 
les systemes de valeurs de 
f 9, h, k, l, 
se reduiront a l’un des sept systemes 
* It would be convenient to say, a term of the cyclical species 2(1) + 1(2) + 1(3). 
