482 Proceedings of Boy al Society of Edinburgh. [july 18, 
a a a a 
a. (3(3 
a a a . 
a a a a 
P/3/3 
Py 
(3(3(3 
P-y 
y 
y 
yy 
y 
8 
8 
quarum numeri sunt 
9 
6 
9 
7 
qui complexionibus datis prsefigi jubent signa 

+ 

5) 
The proof that this rule of signs, which is manifestly nothing else 
than Cramer’s, leads to the same results as the previous rule, is 
quite easily understood if a particular permutation he first con- 
sidered. For example, let the sign of the particular permutation 
8(3ay be wanted. Following the first rule, we should require to 
note four different members, viz., 
(1) the no. of the column in which 8(3ay occurs in the 4th group, 
(2) „ „ 8(3a „ 3rd „ 
(3) „ „ 3/3 „ 2nd „ 
(4) ,, ,, 8 ,, 1st ,, 
The first of these numbers being 1, we should infer that in fixing 
the sign of 8(3ay in the fourth group there had been no change from 
the sign of 8(3a in the third group ; the second number being also 1, 
we should make a like inference ; the third number being 2, we 
should infer that in fixing the sign of 8/3 in the second group there 
had been 1 change from the sign of 8 in the first group ; and 
finally, the fourth number being 4, we should infer that in fixing 
the sign of 8 in the first group there had been 3 changes from the 
sign of a in that group. The total number of changes from the 
sign of a in the first group being thus 3 + 1 + 0 + 0, i.e., 4, the sign 
would be made +. Now the 3 in this aggregate is simply the 
number of letters in the first group which precede 8, the 1 is simply 
the number of letters taken along with 8 before /3 comes to be taken 
along with it to form 8/3 in the second group, and the two zeros 
correspond to the fact that 8(3a on the third group and 8/3ay on the 
fourth group have no permutation standing to the left of them. 
Consequently to count the number of changes (3 + 1 + 0 + 0) from the 
