348 
Proceedings of Royal Society of Edinburgh. [sess. 
which is the next case of the theorem, viz., the case where the 
diagonal elements of the determinants in the development are all 
0 except one. 
Again, adding the first of the C 4n terms to the term before it, 
the first three of the C 4 , 2 terms to the remaining three of the C 4Jl 
terms, the first three of the C 4 , 3 to the remaining three of the C 4 , 2 
terms, and the last term to the remaining one of the C 4 , 3 terms, we 
obtain the next case of the theorem, viz., that used in § 4. 
Treating the expansion now reached in the same way, we find 
the next case, viz., 
1^2 C 3^4 e 5 — 
CL i ^2 CL^ CL^ 
+ 2^4 
CL i ^ CLq 
+ d^e b 
CL | (Lq CL<^ 
b 1 b 2 b B & 4 b 5 
\ \ h h 
b x l> 2 b 3 
C 1 C 2 C 3 C 4 C 5 
C 1 C 2 C 3 C 5 
C 1 C 2 C 3 
6?1 6^2 ^ • ^5 
e i e 2 e 3 * 
e i e 2 e 3 e 4 * 
(E 
and thence, in the same way, 
| I : 
a 1 a 2 a 3 a 4 a 5 
\ h h \ h 
C 1 C 2 C 3 C 4 C 5 
d y dc) d^ d^ d^ 
e i e 2 e 3 e 4 • 
+ e 5 f a 1 a 2 a 3 a 4 
\ \ h h 4 
C 1 C 2 C 3 C 4 
d ^ d '2 d ^ d ^ 
and, lastly, of course, 
| af.gcgl^e^ j = a 1 & 2 c 3 cZ 4 e 5 
(E 5 ) 
6. The reverse process is equally interesting. Starting with 
I a A c 3 ^ 4 e 5 I we separate its terms into two groups, viz., 
(1) those which contain e 5 , 
(2) those which do not, 
'and thus obtain (E 5 ). 
Then we take each of the two groups of (E 5 ) and separate it 
into two groups, viz., 
(1) those which contain d 4 , 
(2) those which do not. 
In this way we obtain the four groups of (E 4 ). 
I^ext we partition these four into the eight of (E 3 ), and so on. 
