570 Proceedings of Royal Society of Edinburgh. [sess* 
Cauchy (1841). 
[Note sur la formation des fonctions alternees qui servent a 
resoudre le probleme de l’elimination. Comptes Rendus . . . 
Paris , xii. pp. 414-426 ; CEuvres, l re ser. vi. pp. 87-99.] 
The early part of this paper, in which the finding of the terms 
of a general determinant (“ fonction alternee ”) is made dependent 
on a study of the properties of “ groups,” or index-cycles as they 
would more appropriately he called, has already been described. 
The nature of it will be readily recalled from the mode of writing 
the expansion of the determinant of the 4 th order, viz., 
^ 11 ^ 22 ^ 33*^44 
- 2 
+ 2 
^11^22^34^43 
a i 2^21*^34^43 
+ 2 
- 2 
a H a 23 a 34 a 42 
^ 12 ^ 23 ^ 34^41 ’ 
where under the last H are included all terms (6 in number) whose 
indices form one quaternary cycle, under the preceding % all terms 
(3 in number) whose indices form two binary cycles, and so on. 
On coming to consider a determinant in which = a # , Cauchy 
points out that because of this peculiarity every term will be found 
repeated unless those whose index-cycles are all lower than ternary r 
for example, in the case of the determinant of the 4 th order, the 
six terms having a quaternary index-cycle are condensed into- 
three with the coefficient 2 prefixed, and the eight terms having 
a ternary index-cycle into four with the same coefficient, the whole 
result being — 
*^11*^22*^33^44 ^11^22^34 "h 2 *hl *^23*^34*^24 
+ , ^13^24 — ^ ^12^23^34^14 ' 
The definite theorem reached by him on this point may be 
formulated in later phraseology as follows : — 
If v 3 , v 4 , . . . be the number of ternary , quaternary , and higher 
index-cycles in any term of an axisymmetric determinant , the co- 
efficient of the term 'when condensation takes place is 2 1/3 + v 4 + • • 
By way of proof it is stated that when we have got a term with 
index-cycles higher than binary, we may, by reversing the order 
of the indices in one of the said cycles, obtain another term of the 
