260 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
[406 
the other set, I consider, for example, the case where the sum of the numbers in the 
{ ), or [ ] is = 5 ; and I form a kind of tree as follows : 
the formation of which is obvious ; and I derive from it in the manner about to be 
explained the expressions for the coefficients [14], [23] &c. in terms of the corresponding 
coefficients in ( ) ; viz. we have 
[14] 
= - 
(14), 
[23] 
= - 
(23), 
[113] 
= 
2 
(14)(13) 
+ 
(23)(11) 
- 
(113), 
[122] 
= 
(14)(22) 
+ 
2 
(23)(12) 
- 
(112), 
[1112] 
= - 
6 
(14) (13) (12) 
- 
3 
(14)(22)(11) 
+ 
3 
(14)(112) 
- 
6 
(23) (12) (11) 
+ 
3 
(113) (12) 
+ 
I 
(23)(111) 
+ 
3 
(122) (11) 
- 
I 
(1112),