590 MR TALBOT ON SOME MATHEMATICAL RESEARCHES. 



In this example the roots 4 and 37 are far from being " approximate," which 

 is the reason why the root was not found at once. 



These problems may be treated in another manner. If x* + qx — r = 0, and 



x, y, z, are the roots, we have seen that — 4q — Zx 2 = D 2 , where D denotes y — z, 



as before. Now, though D is unknown at present, we may make two different 



suppositions concerning it : — First, that it is small; secondly, that it is large, in 



comparison with the root x. If we make the first supposition we may try to 



— 4(7 D 2 



solve the equation, put in the shape — g- — x 2 — -g-, by supposing (since D is 



— 4(7 



small) that x 2 is that square which is nearest to —^ .but smaller than it. This 



supposition is often justified by the result, on trial. But if not, let us make the 

 other supposition, and try to solve the equation (put in the shape — 4q — D 2 

 = 3x 2 ), by supposing (since D is large) that D 2 is that square which is nearest to 

 — 4# but smaller than it. This also very often succeeds and gives at onCe the 

 value of 3x 2 , and thence of x. The first of these methods is essentially the same 

 with that described in the preceding pages, but the second is different. If both 

 of them are employed, I find that one or other of them solves every case of the 

 equation x?+ qx — r — 0, provided the coefficient q does not exceed 200. How 

 much further the success of the method extends I have not yet had leisure to 

 ascertain. I will conclude by giving an example. We find that the equation 

 x i — 1533 x — r — was not soluble by the first method without the help of a 

 " second approximation," but it is readily soluble by the second method, as 

 follows :— 



Since —q = 1533, therefore — 4q = 6132. Assume D 2 to be the greatest 

 square which is less than G132, this will be 6084. Hence — 4q — D 2 = 48. 

 Putting this = 3x 2 we get x 2 — 16, which, being a square number, shows that we 

 are right. Therefore x — 4. And since D 2 = 6084, D = 78. Hence y + z = — 4 

 and y — z = 78, whence y = 37, z = — 41, and x = 4. 



We may therefore draw this conclusion, that " If the equation x 3 + qx — r = 

 has integer roots, and q is a number less than 200. the roots can always be found 

 by the simple extraction of the square root." 



