NUMERICAL EQUATIONS OF ALL DEGREES HAVING INTEGER ROOTS. 307 
reduct is 7. As before, the root will consist of 2 figures. Since the power 
ends in 7, the root ends in 7. It is therefore comprised in the series: 
Of ce Olt ea On HOU mT Or. OT . 
Since the reduct of the power is 7, the reduct of the root is 4. Hence the root 
is comprised in the series: 
13 .22.31.40.49.58.67.76.85.94. 
But the only number which is found in both series is 67. Therefore 67 is the 
root required. 
These theories enable us to effect the solution of numerical equations of any 
degree, whose roots are integers, to an apparently unlimited extent; but I will 
not enter upon that subject in all its generality at present, because from the late 
period of the Session at which this paper was commenced, it has been impossible 
to give it the desired extent and generality. I will confine myself on the present 
occasion to the solution of a particular case which occurs when the roots of the 
given equation are small numbers, less than 97+ 9 = 90. In that case these 
roots can be found, if I am not mistaken, whatever be the degree of the equa- 
tion, by a direct and very simple process. 
My first example shall be a cubic comprised in what is called the Irreducible 
Case. 
| Let the given equation be 
| av —Ax’+ Be—-C=0. 
) Instead of the coefficients write their reducts a, b, c, &c. This gives 
| ; a — an’? + ba—c=0, 
which I call the reduct equation. 
| Let the three roots be a, 2’, x’, and let x be of the form 9y + ». We can 
determine » without knowing y. 7 is one of the nine digits 1.2.3.... 9. 
| Substitute these digits successively for x in the reduct equation, and it will be 
seen which of them (generally three) satisfy that equation, viz., by giving a 
reduct = 0. This is most conveniently done by the help of the little table 
given before. 
TABLE A. Thus, for instance, if we are trying whether 7 = 4, we have 
ow 
1 * i the reduct equation 1 —7a + 4b —c, andif a, b, ¢ are such as 
| 2 #4 8g. to satisfy this by giving a reduct = 0, then we may suppose 4 
| : ; ; to be one of the values of 2. 
| ° ae These things being premised, I find that if the roots are 
| P A : each of them less than 90, the solution reduces itself to the 
/ sg 41. 8. following very simple form. 
| oO 0 Rule—Put «=9y +n, and give n such a value as to satisfy 
the reduct equation 2° — az? + ba —c=0. 
VOL. XXVII. PART III. 41 
