268 
MR. A. CAYLEY ON A NEW AUXILIARY EQUATION IN 
12 
.34 
56 
12 
35 
46 
12 
.36 
.45 
13 
.25 
.46 
13 
24 
56 
13 
.25 
.46 
14 
26 
35 
14 
25 
36 
14 
.23 
. 56 
15 
24 
.36 
15 
26 
34 
15 
26 
34 
16 
23 
45 
16 
23 
45 
16 
.24 
.35 
12 
34 
56 
12 
35 
.46 
12 
36 
45 
13 
26 
45 
13 
26 
45 
13 
24 
56 
14 
25 
36 
14 
23 
56 
14 
26 
35 
15 
23 
46 
15 
24 
36 
15 
23. 
46 
16 
24 
35 
16 
25 
34 
16 
25 
34 
which is in fact the theorem whereon depends the existence, for six letters, of a 6 -valued 
function not symmetrical in respect of five letters. There is not any peculiarity in the 
syntheme of dnads which above presented itself ; the occurrence of this particular syn- 
theme, instead of any other, arises merely from the arbitrary selection of the suffixes of 
the ip’s. 
12. It is hardly necessary to remark that if the pentagon 12345 had been assumed 
negative instead of positive, the only difference would be that the ip’s would have their 
sio-ns reversed. 
O 
13. I proceed now to the calculation of the Auxiliary Equation. As the working is 
rather easier for that form, I shall in the first instance take for the given quintic the 
denumerate form 
(a, h, c, d, e,fjv, 1 )®= 0 . 
Eepresenting, as before, the roots ^ 3 , ^ 4 , of this equation by 1, 2, 3, 4, 5, and 
writing 
12345=12+23+34+45+51, &c. 
(where on the right-hand side 12, 23, &c. stand for x^x. 2 , x^x.^, &c.), we have to find the 
equation 
1 )«= 0 , 
the roots whereof are 
^.=12345-24135, <^,=21435-^13245, 
9,=13425-32145, <p,= 31245-14325, 
(P3=14235-43125, (p6=41325-12435. 
As aheady remarked, the coefficients are seminvariants, and if the equation is in the 
first instance calculated for the particular casey’= 0 , the terms in y* can be separately 
determined. But putting /=0, one of the roots, say 5, becomes =0, and the remain- 
ing roots 1, 2, 3, 4 are the roots of the quartic equation («, c, d, 1)^ = 0. 
14. Writing for shortness 
1234=12+23+34, &c.. 
and putting also 
A=12 + 34, 
B = 13 + 42, 
C = 14+23, 
