466 Proceedings of Royal Society of Edinburgh. [july 18 , 
is then represented by putting a dot in each of the horizontal rows, 
in the first under 3, in the second under 8 , and so on ; so that if 
the rows he taken in order, and the number above each dot read, 
the given permutation is obtained. For the representation of the 
conjugate permutation nothing further is necessary: we obtain it 
at once if we only turn the paper round clockwise until the 
vertical rows are horizontal, and read off in order the numbers 
above the dots. In the next place the number of “ derangements ” 
belonging to the permutation 3, 8 , 5, . . . .is indicated by insert- 
ing a cross in every small square which is to the left of one dot 
and above another ; thus the two crosses in the first horizontal row 
correspond to the two “ derangements ” 32, 31; the six crosses in 
the second horizontal row to the 
six “derangements” 85, 84, 86 , 
81, 87, 82; and so on. Then it J 
is observed that if we turn the ^ 
paper and try to indicate the 4 
“ derangements ” of the conjugate 5 
permutation by inserting a cross 6 
in every small square which is to 7 
the right of one dot and above ^ 
9 
another, we obtain exactly the 
same crosses as before. The signs 
of the two permutations must thus be alike. 
Immediately following this, the 24 permutations of 1, 2, 3, 4 are 
given in a column, each one having opposite it, in a parallel column, 
its conjugate permutation. The existence of self-conjugate permuta- 
tions, e.p., the permutation 3, 4, 1, 2 is thus brought to notice, and 
the substance of the following theorem in regard to them is given : — 
If be the number of self-conjugate permutations of the first n 
integers , then 
U« = U w -i + (w- l)U n _ 2 . .... (xxv.) 
where Uj = 1 and U 2 =* 2 . 
This, however, is the only one of his results which Rothe does not 
attempt to prove. 
In the second part of the memoir, which contains the application 
of the theorems of the first part to the solution of a set of linear 
