576 MR TALBOT ON SOME MATHEMATICAL RESEARCHES. 



We have just found that their sum m + n is equal to — 3r. Let us seek the 

 value of their product mn. 



Multiply m = x 2 y + y 2 z + z 2 x by the first term of n, which is y 2 x, and we get 



x& y s + y 3 • x v z + x?y 2 z 2 



Then multiply by the second and third terms of n, and take the sum of all the 

 results, and we evidently shall have 



mn = Sx 3 y 3 + rSx 3 + 3r 2 



.-. substituting the values already found of 



Sec 3 = 3r and Sx 3 y 3 = q 3 + 3r 2 

 we find mn = q 3 + 9r 2 



Now we have found (m + n) 2 = 9r 2 



and from this subtracting 4mn = 4q 3 + 36r 2 



we get (m — n) 2 = — 4q 3 — 27r 2 



from whence the separate values of m and n follow at once. 



(4.) If the roots of a cubic, written in any order, are a, y, z, the differences of 

 the roots, taken in order, are x — y, y — z, z — x. The other three differences, 

 y — x, &c, are merely the negatives of the three first. 



Theorem. — In any cubic equation, wanting the second term, the product of 

 the three differences of the roots, or, 



± x — y .y — z . z — x = V — (4q 3 + 27i* 2 ) 



For, by actual multiplication, we find 



Z — X 

 y-z 



yz — xy — z* + xz 



x^y 



xyz — x 2 y — xz 2 + % 2 z — y 2 z + xy 2 + yz 2 — xyz . 



Omitting the first and last terms, which destroy each other, the result is 

 (— x 2 y — y 2 z — z 2 x) + (y 2 x + z 2 y + x 2 z), or — Sx 2 y + Sy 2 x, or, according to our 

 previous notation, — m + n. If we take the product of the other three differences 

 of the roots, we get m — n. 



But we have found the value of m — n to be */ — (4q 3 + 27r 2 ) ; therefore the 

 theorem is proved. Several important consequences follow. In the first place, 

 if the roots are whole numbers, their differences are so ; whence this theorem. 



"If the roots of the cubic x"° + qx — r = are whole numbers, the quantity 

 4q 3 + 27 V 2 is necessarily the negative of a square." This theorem is, I believe, 

 due to Legendre. To give a few examples of it — 



