422 Proceedings of Royal Society of Edinburgh. [sess. 
that part of the term which corresponds to one of the groups of 
indices. In the case of the group (3), the number of interchanges 
s 0; in the case of the binary group (0, 1) it is 1 ; and in the case 
of the quaternary group it is 3 — the number of interchanges being 
“evidemment ” one less than the number of indices in the group. 
If, therefore, for a given term there be in all m groups, viz. / groups 
of one index each, g groups of two indices each, h of three, k of 
four, &c., the number of necessary interchanges will be 
Of + l.g + 2 .h + 3 .k + 
which 
f J r 2 . g + 3.7?/ + i.k + . . . , 
— (/ + g + h + k +...), 
f--n- to ; 
and consequently the sign of the term will be + or — 1 according 
as n - m is even or odd. (hi. 28) 
The first step of the new investigation is to define “ termes 
semblables ou de meme esp&ce.” Two terms are said to be alike or 
of the same species when the one may be obtained from the other by 
subjecting both sets of indices in the latter to one and the same sub- 
stitution or permutation. Thus recurring to the term above used, 
tt 0.1^1.0 a 2.5%.3 tt 4.6 ffl 5, 4^6.2 > 
and substituting in both of its sets of indices 6, 0, 1, 4, 3, 2, 5, 
instead of 0, 1, 2, 3, 4, 5, 6 respectively, — in other words, and with 
the notation of the memoir of 1812, performing the substitution 
/0 1 2 3 4 5 6\ 
\6 0 1 4 3 2 5/, 
we obtain the like term 
tt 6,0 a 0.6^1.2 a 4>4 a 3,5 a '2.3 a 5.1 • ( LV ‘) 
The groups in two like terms are evidently similar, the values of 
f g , h, . . . for the one being the same as those for the other. 
Indeed, since it is in this matter of groups or cycles that the terms 
have any likeness at all, the expression “ cyclically alike ” would 
have been a better term for Cauchy to use. 
From the definition there arises the self-evident proposition — 
Terms which are cyclically alike have the same sign. (hi. 29) 
