om Samed fh fee fed famed fame fore famed 
e e ° ° e. ° . . e 
DD QD Sr Or Or 
CD OWOONOON 
518 Prof. Sylvester on a Four-valued Function 
1.7.4 2.97235 5.5.6 
127.25 92.9.) 3.8.4 
1.7 .6_-2.9 435 48..p 
1.8.4" 27 uate), D 
128.6282 .7 0: 73.9.4: 
1.8.0 2a ok ps aD 
1 ls eee: eee 
1.9.06 -2.8.50.3.%.4 
19.09. 2.6. Od 20 
The whole of this family of 36 may be represented under 
the following condensed form, according to the notation usual 
in the theory of substitutions. 
123 123 .1238 . oct! ax aa bB °”) 
123 281 312 cn! oly aa aa ae 
It remains to describe the principal and most symmetrical 
family. This contains only 4 groupings, and may be repre- 
sented indifferently under any of the three following forms: 
ay 
a 
a\x 
As & 
as 
aa aa aa aa 1,8 b,f' 6,8 5, ay ey ey ey 
Aint Age! det doe! or 6,8 5,6! 5,8 6/3! or Coy Cay! Coty Cony! 
age dget! age age 058 0,8! 6,8 6,8! — egy egy! Cay Cty’ 
In developing, it will be found that each of these three represen- 
tations gives rise to the same family of groupings, which from 
its importance it is proper to set out in full as follows :— 
2.5.8 3.6.9|/1.4.7 2.5.9 3.6.8/1.4.7 2.6.8 3.5.9|1.4.7 2.6.9 3.5% 
2.5.9 3.6.7/1.4.9 2.5.8 3.6.7|1.4.8 2.6.9 3.5.7|1.4.9 2.6.8-8.5 
2.9.7 3.6.8/1.4.8 2.5.7.3.6.911.4.9 2.6.7 3.5.8/1.4.8 2.627 S.b5 
2.6:8 3.4.9/1.5.7 2.6.9 3.4.8/1:5.7 2.4/8 3.6.9|1.5.7 2.4.59 3.6 
2.6.9 3.4.7/1.5.9 2.6.8 3.4.7/1.5.8 2.4.9 3.6.7|1.5.9 2.4.8 3.6. 
2.6.7 3.4.8/1.5.8 2.6.7 3.4.9)1.5.9 2.4.7 3.6.8)/1.5.8 2.4.7 3.6 
2.4.8 3.5.911.6.7 2.4.9 3.5.8|1.6.7 2.5.8 3.4.9|1.6.7 2.5,.950.45 
2.4.9 3.5.7/1.6.9 2.4.8 3.5.7/1.6.8 2.5.9 3.4.7/1.659 Jane 
2.4.7 3.5.8/1.6.8 2.4.7 3.5.9/1.6.9 2.5.7 3.4.8|1.6.8 2.5.7 3.4 
OHO WNI OD CONT 
It bios at once from the above Table, that if 3 cubic equa- 
tions be given, we may form a function of the 9 roots, which, 
when any of the roots of any of the equations are interchanged 
inter se, or all the roots of one with all those of any other, will 
receive only four distinct values. . 
It also follows that we may form with 9 letters an intransitive 
group (of Cauchy) containing ™°, i. e. 54, or a transitive group 
