140 
PROFESSOR CAYLEY ON THE CURVES 
abcd= abed 
+ [13](a . bcd+ &c.) 
+ [22](ab.cd+ &c.) 
+ [112](a.b.cd+&c.) 
+[1111] a.b.c.d, 
and so on; and it is required to find the relation between the coefficients ( ) and [ ] ; 
we find, for example, 
[11]=- 
(11), 
[12]=- 
(12), 
[Hl]= 3 
(11)(12) 
- 
(HI), 
[13]=- 
(13), 
[22]=- 
(22), 
[112]= 2 
(13)(12) 
+ 
(22)(11) 
- 
(H2), 
[1111]=- 12 
(13)(12)(11) 
+ 4 
(13)(1H) 
- 3 
(22)(11)(11) 
+ 6 
(112)(H) 
- 
(1111). 
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)= + I2[13][12][11] 
= (12) — 12+12 =)— 12 
[13][12][11] 
- 4[1S]{3[12][11]-[111]} 
+ 4 
[13][111] 
+ 3[22][11][11] 
+ (3-6= )- 3 
[22][11][11] 
- 6[11]| 2[1S][12]| 
+ 6 
[11 2] [11] 
+[22][11] 
- 
[HU] 
— [mi]l-[112] J 
- 
having the same coefficients —12, +4, 
— 3, +6, —1 as in the formula for [1111] 
terms of the coefficients ( ) ; it is easy to infer that the property hold goods generally. 
To explain the law for the expression of the coefficients of either set in terms of the 
other set, 1 consider, for example, the case where the sum of the numbers in the ( ), or 
