346 
Proceedings of Royal Society of Edinburgh. [sess. 
But the terms containing all three are known to be 
C 3^4 e 5 
a , 
a. 
The terms containing only two are those containing only c 3 d± , 
c 3 e 5 or <i 4 e 5 ; and as all those containing c 3 d ± , c 3 e 5 , d^e 3 are re- 
spectively 
*A\ ^ 1 ^ 2 e 5 ^ J C 2 e b\ a ffl± | , ^465] af 2 C 3 | , 
those containing only c 3 d ± , c 3 e 5 , d A e 5 are respectively 
C 3^4 
CL 1 ^2 
b i h h b 
C Z e b 
a x a 2 « 4 
^1 ^2 ^4 
rf 4 e 5 
6^2 ^3 
6 4 5 2 & 3 
e i e 2 
i 
J 
C 1 C 2 
Similarly, the terms containing only one are those containing only 
c 3 , d± or e 5 ; and as all those containing c 3 , d± or e 5 are respec- 
tively 
C 3 i tt 1^2^4 e 5 I > ^4 I a f > 2 C Z e b I > e 5 I tt 1^2 C 3^4 
those containing only c 3 , d±, e 5 are respectively 
«1 
a 2 
« 4 
Cl 5 
j d^ 
a Y 
a 2 
<h 
a b 
» e 5 
a x 
a 2 
a 3 
« 4 
\ 
\ 
^4 
h 
\ 
b ‘2 
h 
h 
\ 
*2 
h 
^4 
d l 
d 2 
• 
^5 
c i 
c 2 
• 
C 5 
e i 
C 2 
• 
C 4 
e i 
e 2 
e 4 
. 1 
e i 
e 2 
e 3 
• 
d 1 
d 2 
d 3 
. 
Lastly, the terms containing none of the three selected elements 
are 
CL-^ $2 
b 1 b 2 b 3 b 4 
c i c 2 • C 4 
d-^ {^2 c? 3 * 
e 1 e 2 e 3 e 4 
Consequently we have the identity — 
CL ^ 
+ 2^ 
CL ?) 
6 2 & 3 5 4 
^1 ^2 ^4 ^5 
C 1 C 2 • C 4 ^5 
6?2 0 ^ 
6^2 ^2 ^3 «• ^^5 
<?i e 2 e 4 
e i e 2 e 3 e 4 
+ 2^4 
a i 
a 2 
<% 
" 1 “ £ 3^465 ^2 
h 
1 & 2 
*1 
e 2 . 
