312 H. F. TALBOT ON A GENERAL SOLUTION OF NUMERICAL EQUATIONS, ETC. 
We now proceed, as in the case of the cubic, to substitute the numbers 
1.2.3.4.5 successively in the given or original equation. The results ¢, 7, 9; 99; 
will all be divisible by nine. Let them be so divided, and the reducts of the 
quotients taken and called respectively Q, Q, Q, Q, Q, . 
separ Ue le Ma gl re cate 
One ; : 2679600 JG, 
Two : : 2227680 Ua 
Three Y : 1836768 3 =O. 
Four : : 1500720 K=30,, 
Five ; : 1213800 6= Q, 

Having now found the values of P and Q for each root, we proceed, as in the 
case of the cubic, to solve the equation Py = Q, and having found y, 9y + » 
will be the root of the given equation. 
First Root.—Here P, = 6 and Q, = 3, therefore 6y = 3 whence y = 2, and 
a = 9y + 1 = 19, which is the root required. 
Second Root.—Here P, = 3 and Q, = 0 .. 3y = 0, which gives y = either 
3 or 6, whence 2 = 9y + 2 is either 29 or 56. This ambiguity will be removed 
hereafter. 
Third Root.—Here P, = 4 and Q, = 3, ..4y =3 whence y =38 and 
a = 9y + 3 = 30, which is the root required. 
Fourth Root.—Here P,=3 and Q,=6, ..3y=6 whencey=2 and 
x = 9y + 4 = 22, which is the root required. 
Fifth Root.—Here P, = 6 and Q, = 6 ... 6y = 6 which gives y = either 1, 4, 
or 7, and 2 = 9y + 5 is either 14, 41, or 68. 
But it cannot be 68, because no root of the given equation can end in 8.* 
Therefore this last root is either 14 or 41. 
Thus we have determined three roots of the given equation to be 19, 30, 
and 22. This sum is 71, and since the coefficient of the second term of the 
given equation is 168, the sum of a// the roots must be 168. Subtracting 71, 
the sum of the known roots, we have 168 — 71 or 97 as the sum of the two 
roots which are not yet determined. But we know that they are some two of 
the four numbers 29, 56, 14, 41, and the only two of these numbers whose sum 
is 97 are 56 and 41. This simple consideration dispels the ambiguity of the 
first calculation, and we conclude finally that the five roots are 19.56.30.22.41. 
I propose to demonstrate the convenient rule by which these equations 
have been solved in the next part of this memoir. 
* This appears by a simple process, but too long to explain here. 
