584 
PROFESSOR CAYLEY ON THE CONDITIONS EOR THE EXISTENCE 
15. Assuming the existence of these relations, we have for the determination of the 
numerical coefficients in each relation a set of linear equations, which are shown by the 
following Tables, viz. referring to the Table headed c9l, 61$, «C, «.1234, if the multipliers 
of terms respectively be A, B, C, X, then the Table denotes the system of linear equations 
0 A + 3B + 33C + 0 X=0, 
3 A + 0B -102 C —16 X=0, 
&c., 
that is, nine equations to be satisfied by the ratios of the coefficients A, B, C, X, and which 
are in fact satisfied by the values at the foot of the Table, viz. 
A:B:C:X=66: -11: +1: +6. 
There would be in all fourteen Tables, but as those for the second seven would be at 
once deducible by symmetry from the first seven, I have only written down the seven 
Tables ; the solutions for the first and second Tables were obtained without difficulty, 
but that for the third Table was so laborious to calculate, and contains such extraordinarily 
high numbers, that I did not proceed with the calculation, and it is accordingly only the 
first, second, and third Tables which have at the foot of them respectively the solutions 
of the linear equations. 
16. The results given by these three Tables are, of course, 
66cg- 1161$+ ltfC + 6«.1234=0, 
330dg[+110d$— 555C+9a3B-105 «.1235=0, 
+266478575 eg 
-617359490 ffi 
+ 144200810 c€ 
+ 9656911 6B 
+ 9090785 a(& 
-721004050 c . 1234 
+ 90914175 6.1235 
-160758675 a. 1245 
+ 11559295 a. 1236=0. 
It is to be noticed that the nine coefficients of this last equation were obtained from, 
and that they actually satisfy, a system of fourteen linear equations ; so that the cor- 
rectness of the result is hereby verified. 
17. The seven Tables are 
