356 
Proceedings of Royal Society of Edinburgh. [sess. 
And, generally, the number of unlike terms in a skew determinant 
of the ni\\ order is * 
1 + c„, + c„ . + 
3-5 -7(1 + 3- 5-7) , 
i ~ "r 
15. If we specialise further, by making m 1 = m 2 = m 3 = m 4 = m 5 = 1, 
the expansion takes the form of a sum of squares. 
If, on the other hand, we make each of the m’s equal to 0, the 
expansion in the case of odd orders entirely disappears ; and, in the 
case of even orders, reduces to one term. This is the “ very special 
case” referred to in §2, and used in §7 in proving the general 
theorem. It is the subject of Cayley’s fourth paper on Skew 
Determinants.! 
It would he well, however, to combine these two special cases 
and others in one statement, viz., that, if the values of the diagonal 
elements of a shew determinant be confined to 0 or 1, the deter- 
minant is expressible as a sum of squares of Pfajfians. For ex- 
ample, in the case of the 5th order, if m x = 0, and m 2 = m 3 = m 4 = 
m 5 = 1, we have 
K 
h 2 
h 
K 
— | h 2 h 3 7q 
2 + | 7q h 3 \ 
2 + 
1 7q h 2 \ 
2 + 
1 b \ h 2 h 3 
\ i 
a i 
a 2 
a 3 
Pi P 2 
a 2 a 3 
a l a 3 
a l a 2 
h 2 - oq 
1 
Pi 
P 2 
7i 
7l 
ft 
Pi 
h$ ~ «2 
- Pi 
1 
7i 
+ hfi + a 3 
— a 3 
~ P 2 
~7i 
1 
16. Before proceeding to our next theorem, the nature and 
notation of the elements of product-determinants require to be 
recalled to mind. The elements of the determinant which is the 
product of | afi 2 c 3 | and | a 1 /3 2 y 3 | are of the form, 
Gqcq “I - a.-)S\ ”1” $ 3 y 4 , 
or (^]& 2 ^ 3 )( a i/^iyi)> 
* Cf. Cunningham’s paper, above referred to, p. 225. 
f Cayley, A. , “ Theoreme sur les determinants gauches, ” Crelle's Journ . , Iv. 
pp. 277, 278 ; or Collected Math. Papers, iv. pp. 72, 73. 
