430 
Proceedings of Royal Society of Edinburgh. [sess. 
be remembered, was to speak of “ applying a substitution to a per- 
mutation.”* 
Of the proposition (b) a proof is given, which may be paraphrased 
as follows : — Let the three permutations referred to change P, the 
original product of differences, into e jP, e 2 P, e 3 P, respectively, the 
e’s of course being either +1 or - 1. Then as the performance of 
the first two permutations in succession will result in the change 
of P into e 1 .e 2 P, we must have 
e i • e 2 = e 3 > 
so that e 1 and e 3 have the same or opposite signs according as 
e 2 is + 1 or - 1 ; and this is virtually the proposition to be 
proved. (in. 30). 
A demonstration of ( d ) is also given. The two permutations 
being A and B, l the first index of A, and m. the first index of B, 
the performance of A on B implies that the I th index in B is to take 
the first place, and the performance of B on A that the m th index of 
A is to take the first place. The resulting permutations will con- 
sequently not agree in the first index, unless the Z th index of B is the 
same as the m th index of A, which manifestly need not be the case, f 
To prove (/) is of course the same as to prove that the interchange 
of two indices r and s, r being the greater, alters the sign of the 
product of differences ; and this is done by separating the product 
into three portions, viz., (1) the portion which contains neither 
a r nor a s ; (2) the single factor which contains both, a r - a s ; and (3) 
the product of all the factors having either one or the other for a 
term. It is then asserted that the interchange of r and s cannot 
alter the last of these, because it is symmetrical with respect to a r 
and a s ; also, that no alteration is possible in the first, and conse- 
quently that the change in the second accounts for the validity of 
the proposition. (in. 32) 
'* He says, for example {Jour, de Vfic. Polyt., x. p. 10), “ Si en appliquant 
suceessivement a la permutation A, les deux substitutions ) et 011 
obtient pour resultat la permutation A 6 ; la substitution^ 1 ^ sera equivalente 
au produit des deux autres et j’ indiquerai cette equivalence 1 comme il suit 
+ This also is a paraphrase of Jacobi’s proof. 
