409] OF EQUAL HOOTS OF A BINARY QUARTIC OR QUINTIC. 307 
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 c2l, 633, a®, a. 1234, [first of 
the seven tables infra\ if the multipliers of the several terms respectively be A, B, C, X, 
then the Table denotes the system of linear equations 
0 A +3 B +33(7 + 01 = 0, 
3 A +0 B -102(7 -16X = 0, 
&c., 
that is, nine equations to be satisfied by the ratios of the coefficients A, B, (7, X, 
and which are in fact satisfied by the values at the foot of the Table, viz. 
A : B : C : X = + 66 : —11 :+l :+6. 
There would be in ail 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, 
66 c2l — 11633+ 1 a® + 6a. 1234 = 0, 
330 + 110 c33 — 55 6g + 9 aS) - 105 a. 1235 = 0, 
+ 266478575 eA 
- 617359490 d33 
+ 144200810 cS 
+ 9656911 65) 
+ 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 
correctness of the result is hereby verified. 
39—2