1887.] Dr T. Mair on the Theory of Determinants. 
465 
and the two series of signs are seen to be identical, 3 being an odd 
number. Viewing this quite independently of the theorem to 
which it is annexed, it is evident that a change of sign at any 
point in the series (A) implies a change at the corresponding point 
in the other series, and consequently attention need only be paid to 
the first sign of (B) as compared with the first sign of (A). Now 
the first sign of (A) must necessarily be always plus, there being no 
inversions; and the first sign of (B) depends on the changes 
necessary for the transformation of the natural order 1, 2, 3, 4, 
into 3, 1, 2, 4. The truth of the corollary is thus apparent. 
A second corollary is given, but it is of still less consequence, the 
difference between it and the first being that in the arranged set (B) 
the place whose occupant remains unchanged may be any one of the 
n places. (hi. 12.) 
The next few paragraphs concern the subject of “ conjugate 
permutations” ( verwandte Permutationen ), — apparently a fresh 
conception. The definition is — 
Tivo permutations of the numbers 1, 2, 3, . . . , n are called 
conjugate when each number and the number of the place which it 
occupies in the one permutation are interchanged in the case of the 
other per mutati on. (xxiv.) 
For example, the permutations 
3, 8, 5, 10, 9, 4, 6, 1, 7, 2 (A) 
8, 10, 1, 6, 3, 7, 9, 2, 5, 4 (B) 
are conjugate, because 3 is in the 1 st place of (A) and 1 is in the 
3 rd place of (B), 8 is in the 2 nd place of (A), and 2 is in the 8 th place 
of B, and so on in every case. 
The first theorem obtained is — 
Conjugate permutations have the same sign. (hi. 13.) 
This is proved in a curious and interesting way, a special 
conjugate pair being considered, viz., the pair just given as an 
example. To commence with, a square divided into 10 x 10 equal 
squares is drawn, the vertical rows of small squares being numbered 
1, 2, 3, &c. from left to right, and the horizontal rows 1, 2, 3, &c. 
from the top downwards. The permutation 
3, 8, 5, 10, 9, 4, 6, 1, 7, 2 
