134 
Proceediifigs of Royal Society of Edinburgh. 
A Relation between Permanents and Determinants. 
By Thomas Muir, LL.D. 
(Read December 20, 1897.) 
1. If all the negative terms of the determinant | af) 2 c 3 ... | be 
changed in sign, we obtain a symmetric function, dealt with by 
Borchardt and Cayley, known as a Permanent and denoted by 
The more important elementary properties of such functions are 
given in a paper published in the Proc. Roy. Soc. Edin., xi. 
pp. 409-418. As might be expected, relations are found to exist 
between them and determinants, an important instance being the 
theorem of § 7 of the said paper. Another theorem, not hitherto 
noted, deserves now to be put on record. 
2. For the case of the 2nd order it is 
which is easily verified by observing that the coefficients of a v a 2 , 
a 3 in the expression on the left-hand side are respectively 
I a f2 C S 
i«AI - “A ( 
+ r — 0. 
the truth of it being self-evident. 
For the case of the 3rd order it is 
— | a x b 2 c z j 4 - a 1 j b 2 c 3 j + b 2 \ a l c 3 | 4- c 3 j a 1 b 2 | 
+ + + +' + + + + Y” 
+ af 2 c 3 ~ V 1 — ^ 2 ! ^ 1 ^ 3 ! — ^ 3 ! J 
- | V 3 I + I & 2 C 3 i + V*3 + Vs ) 
i V 3 i ” Vs "b i V 3 I Vs> 
+ + 
I Vl i “ Vl "b i \ C 2 i Vl 3 
and that by the previous case each of these vanishes, 
