410 
Proceedings of the Royal Society 
of elements. It was in this light that determinants were viewed 
and treated of by Cauchy. 
On the other hand, if all the negative signs of a determinant be 
made positive we obtain a permanent symmetric function with re- 
spect to the same groups of the same quantities. It is functions of 
this kind, resembling determinants, which we wish to consider in 
the present paper. Two notations have before this been used for 
them : the above instance may be specified, after Cauchy, by 
%{af 2 c^)' l 
or, after Cayley, by 
As however the contraction, {af) 2 c 3 ) for the latter form is not 
sufficiently distinctive, it would be better perhaps to use instead the 
notation 
C 1 C 2 C 3 
which is very conveniently contracted into 
+ + 
I a f > 2 C 3 I* 
Tor shortness, in speaking of the functions, we may agree tem- 
porarily to call them Permanents. 
+ + . 
3. The Permanent | a 1 b 2 c 3 | is evidently equal to 
( a \i i T d - eifii) (bfi + + bff) {pf\ + ef 2 *h ? 
if iff 3 = 1 and i 1? i 2 , i 3 be symbols subject to the laws of ordinary 
algebra, except that i\ = i\ — i\ = 0. 
+ d 
It is almost the same to say that | a 1 b 2 c s \ is the coefficient of xyz 
in the expansion of the product 
(a x x + a 2 y + af) (b-^x + b 2 y + b 3 z) (c^ + c 2 y + cf) . 
