406] 
ON THE CURVES WHICH SATISFY GIVEN CONDITIONS. 
259 
and so on; and it is required to find the relation between the coefficients ( ) and 
[ 1 ; we find, for example, 
[11] = - 
(11). 
[12] = - 
(12), 
[HI] = 
3 
(11)(12) 
- 
(HI), 
[13] = - 
(13), 
[22] = - 
(22), 
[112] = 
2 
(13)(12) 
+ 
(22)(11) 
- 
(112), 
[1111] = - 
12 
(13)(12)(11) 
+ 
4 
(13)(111) 
- 
3 
(22)(11)(11) 
+ 
6 
(112)(11) 
- 
(mi); 
and it is to be noticed that, conversely, the coefficients ( ) are given in terms of the 
coefficients [ ] by the like equations with the very same numerical coefficients ; in 
fact from the last set of equations, this is at once seen to be the case as far as 
(112); and for the next term (1111) we have 
(1111) = +12 [13] [12] [11] 
- 4 [13] {3 [12] [11] -[111]} 
+ 3 [22] [11] [11] 
- 6 [11] ( 2 [13] [12]] 
+ [22] [11] 
- [1111] l - [112] J 
having the same coefficients — 12, +4, — 
in terms of the coefficients ( ); it is € 
generally. 
= (12 - 12 — 12 =) - 12 [13] [12] [11] 
+ 4 [13] [111] 
+ (3-6= )- 3 [22] [11] [11] 
+ 6 [112] [11] 
- [HU] 
, +6, — 1 as in the formula for [1111] 
sy to infer that the property holds good 
To explain the law for the expression of the coefficients of either set in terms of 
33—2