184 
Proceedings of Royal Society of Edinburgh. [sess. 
It is then pointed out that when the first coefficient has been got 
in one of the numerators, the others are arrived at by circular 
permutation, the elements permuted being 12345 in the case of 
the first numerator, 02345 in the case of the second, 01345 in the 
case of the third, and so on ; also that the first coefficient in one 
line is got from the last in the preceding line by changing 012345 
into 123450 and then transposing the first two elements ; and that 
these laws hold generally. 
A general mode of finding the ordinary expression for the new 
functions here introduced and symbolized by 
(1234), (123456), 
is next explained. It is first stated that the number of terms 
represented by 
(2,3,4, ...,p) 
where p is necessarily an odd integer is 
1.3.5 (p-2), 
and that one of them is 
(28).(46).(67) (p- l,jp). 
We are then told to permute cyclically the last p - 2 elements 
3,4,5, . . . , p, and we shall obtain from this p — 2 terms in all ; 
thereafter to permute cyclically the lasty? - 4 elements 5,6,7, . . . p 
in each of they? - 2 terms just obtained, and so on. For example, 
(234567)= (23)(45)(67) + (23)(46)(75) + (23)(47)(56) 
+ (24)(56)(73) + (24)(57)(36) + (24)(53)(67) 
+ (25)(67)(34) + (25)(63)(47) + (25)(64)(73) 
+ (26)(73)(45) + (26)(74)(53) + (26)(75)(34) 
+ (27)(34)(56) + (27)(35)(64) + (27)(36)(45). 
It is important to note in conclusion, that the case of an odd 
number of equations is not neglected by Jacobi, a proof being given 
by him that in that case the determinant of the system vanishes. 
In his own words — which are interesting in view of what has 
been said elsewhere regarding the evidence which the paper affords 
of the progress made by him in the study of determinants — 
“ Hun bleibt nach dem bekannten Algorithmus, nach 
welchem die Determinante gebildet wird, diese unverandert, 
