CUBIC EQUATIONS. 585 



Example 1 . Let the roots be 3, 7, — 10. Then q = 21 — 70 — 30 = — 79, 

 .-. — 4q = 316. Since the values of x are 3, 7, — 10, the values of x 2 are 

 9, 49, 1 00, and those of Sx 2 are 27, 147, 300. Hence the values of — 4q — Sx 2 are 

 289, 169, 16, which are all square numbers. 



Example 2. Let the roots be #=11, #=13, z— — 24, whence — 4g = 1732. 

 Since the values of x are 11, 13, — 24, those of — 4q — Sx 2 are — 



1732 - 363 = 1369 = 37 2 

 1732 - 507 = 1225 = 35 2 

 1732 - 1728 = 4 = 2 2 . 



Extracting the square roots, and taking the three roots x, y, z, cyclically, or in 

 regular order, we have 



y - z = 37 



z — x — — 35 

 x - y = - 2 . 



The sum of these three equations gives = 0, since x + y + z = 0hy hypothesis. 



I come now to the principal object of this paper, which is to inquire under 

 what circumstances, or in what cases, the value of one of the roots can be found, 

 since the two others immediately follow. We have seen that — 4q — Sx 2 is neces- 

 sarily a square, therefore, in the first place, x 2 must be such a square as not to 



exceed ^^ ; and, in the next place, whatever square N 2 is tried, it is necessary 



that 3N 2 , subtracted from — 4g, should leave a square remainder. But if it were 

 necessary to try all the squares N 2 which answer the first condition, that of not 



exceeding ~~ g , such a process would be impracticable from its length (except in 



equations with small coefficients). 



While meditating upon this subject, I have met with a theorem which appears 

 to me rather of a novel kind, and which may perhaps open some new views in 

 the theory of equations. First, I must define what I understand by " Approximate 

 roots." If two roots x, y, of the equation x z + qx — r = which has integer 

 roots, are so nearly equal (regard being paid to their magnitude) that 2 (x + y) + 1 



is greater than ^ g , then I call them " approximate roots." Such numbers, 

 for example, are 17 and 34, because 2 (x + y) + 1 = 103, which is greater than 

 \- x ~~y' or 96^. But 17 and 35 are not " approximate numbers, 1 ' because 

 2 (x + y) + I = 105, which is less than * x ~~¥' or 108. It does not, therefore, 



follow that the numbers are at all nearly equal, because they are approximate. 

 Indeed, the difference between them may be very large, provided the numbers 

 themselves are both of them large. 



VOL. XXIV. PART III. 7 T 



