354 Proceedings of Royal Society of Edinburgh. [sess. 
of 9. Similarly, when we come to the 7th order the square of a 
15-termed expression turns up, and this we must now count as 
containing, not 15 2 terms, hut 15 2 -~C 15 , 2 . The numbers of unlike 
terms are thus — 
for the 5th order 1 + 4*1 + 6*3 + 4'3 2 + (3 2 *5 - 3) = 101, 
„ 6th „ 1 + 5*1 + 10'3 + 10‘3 2 + 5(3 2 *5 - 3) + (15 2 - 5*3) = 546, 
„ 7th ,, 1+6*1 + 15 *3 + 20*3 2 + 15(3 2 ’5 - 3) + 6(15 2 - 5'3) + (15 2 *7 - 105 - 6 *5 *3) = 3502 , 
12. As a bordered skew determinant of the rath order can be 
expressed as a sum of n - 1 such determinants of the (ra - l)th 
order, together with a skew determinant of the latter order, and as 
the number of unlike terms in a skew determinant is known,* it is 
clear that we have a ready means of verifying the figures just 
obtained. 
Denoting the number of unlike terms in a skew determinant of 
the rath order by S n we have * 
S 3 = 4 , S 4 = 13 , S 5 = 41 , S 6 = 226 , S 7 = 1072, . . . 
so that, if (BS) n be used in a similar way in connection with 
bordered skew determinants, we have 
(BS) 4 = S 3 + 3(BS) 3 = 4 + 3-6 = 22, 
(BS) 5 = S 4 + 4(BS) 4 = 13 + 4-22 == 101, 
(BS) 6 = S 5 + 5(BS) 5 = 41 +5-101 = 546, 
(BS) 7 = S 6 + 6(BS) 6 = 226 + 6-546 = 3502 , 
exactly as in the preceding section. 
It should also be noted that as a consequence of this 
(BS) g = S 7 + 7Sg + 7-6S 5 + . . . . + 7-6-5.4-3-2-1 . 
13.- The most important special case of the theorem is got by 
putting ~k x , k 2 , k 3 , . . . = \ , h 2 , h 3 , .... , for then the bordered 
determinant becomes itself a skew determinant. A notable change 
takes place also in the development, the two Pfaffians in every pro- 
duct, even where in the general theorem they are of different 
* Cunningham, Allan, “ An Investigation of the Number of Constituents, 
Elements, and Minors of a Determinant,” Journ. of Science, iv. (1874), pp. 
212-228. 
