142 
PKOFESSOK CAYLEY ON THE CUEVES 
+ 30 (23)(12)(11)(11) 
-10 (23)(111)(11) 
-30 (113)(12)(11) 
+ 10 (113)(111) 
-15 (122)(11)(11) 
+ 10 ( 1112 )( 11 ) 
- 1 (11111). 
To form the symbolic parts, we follow each branch of the tree to each point of its 
course : thus from the branch 113 we have 
(113) belonging to 
[113], 
( 11 S)( 111 ) 
[ 11111 ], 
(113X12) 
[ 1112 ], 
(113)(12)(11) „ 
[mu]; 
viz. (113) belongs to [113]; (113)(111), read 11(3 replaced by)lll, belongs to [11111]; 
(113)(12), read 11(3 replaced by) 12, belongs to 1112 ; (113)(12)(11), read 11(3 replaced 
by) 1(2 replaced by) 11 , belongs to [ 11111 ]. 
And observe that where (as, for example, with the symbol 122) there are branches 
derived from two or more figures, we pursue each such branch separately, and also all 
or any of them simultaneously to every point in the course of such branch or branches ; 
thus for the branch 122 we have 
( 122 ) 
belonging to [ 122 ], 
( 122 )( 11 ) 
( 122 )( 11 ) 
• (same twice) 
»’ [ 1 H 2 ], 
( 122 )( 11 )( 11 ) 
„ [ 11111 ]. 
Similarly for the branch 23 we have 
(23) 
belonging to [23], 
(23)(111) 
„ [ 1 H 2 ], 
(23)(12) 
„ ’[ 122 ], 
(23)(12)(11) (same as infra ) 
[ 1112 ], 
(23)(11)(111) 
„ [ 11111 ], 
(23)(11)(12) (same as supra) 
[ 1 H 2 ], 
(23)(11)(12)(11) 
[ 11111 ]. 
We thus obtain the symbolic parts of the several expressions for [14], [23] [11111] 
respectively : the sign of each term is + or — according as the number of factors in ( ) 
