586 ME, TALBOT ON SOME MATHEMATICAL RESEARCHES. 



This definition of " approximate roots " having been given, my theorem is the 

 following : — 



" If an equation x 3 + qx — r = has integer roots, and if two of them are 



approximate roots, then the square root of the quantity — ^-^, rejecting decimals, 



expresses one of the roots of the equation accurately.' 1 '' 



The character of this theorem is like those to which I have already adverted, 

 contained in books of Algebra. " If an equation has two equal roots ; or if it has 

 two equal roots with opposite signs ; or if it has three roots in arithmetical pro- 

 gression, these roots can be found." It is not obvious that any given equation 

 has these peculiarities ; the theorems which I have cited only say that if it hare 

 either of them, the roots can be found. So my theorem only says, "Approximate 

 roots, if they exist, can be found." Only that it has this advantage, that, far 

 from requiring equal roots, which can only seldom occur, it suffices that the roots 

 should be ejusdem generis so to speak, or, in other words, not very unequal, regard 

 being had to their magnitude. 



I will first give a few numerical examples, and then investigate the theory of 

 the subject. 



Example 1. Let x 3 - 2023x + 29478 = 0. 



Here q = - 2023 .-. ^2 - 2697 \ 



the square root of which is rb 51 + some decimals. It will be shown hereafter 

 that we should choose the negative sign ; therefore, rejecting the decimals, try it* 

 — 51 is the root, and it will be found to succeed. The work stands as follows: — 



2023 x 51 = 103173 - 51 s = - 132651 



29478 



132651 



Now to find the other two roots. We have shown that — 4q — Sx 2 is a square 



== R 2 , and that the root x — — 51, and that y = — „ — > z — ~ j~ 



To compute the value of R. We have x 2 = 51 2 = 2601. 



We have also - 4g = 8092 

 3x 2 = 7803 



Difference = 289 



Hence R = \Z289 = 17. And therefore y = 17 t 51 = 34, and z = " ' 17 + ** 

 — 17 .*. the three roots are — 51, 34, and 17. 



The success of the process is owing to 17 and 34 being approximate numbers, 

 although one of them is double of the other. 



Example 2. Let x"° - 73x + 72 = 0. 



Here q= -73.-. ^ii = 97^ 



3 3 



