for determining the Roots of Numerical Equations. 443 



one condition is, that the product of the squared differences 

 shall be a perfect square; in other words, the equation cannot 

 have all its roots rational unless 



p^q^—^<f—lSpqr—^p^r — 2^(r'^ 

 be a positive square number. 



This remark is made at the end of the second supplement 

 of Legendre's Theory of Numbers, and is indeed self-evident ; 

 and in like manner one condition may be obtained for an equa- 

 tion of any degree which is to have all its roots rational ; but 

 this is far from being the sole condition required. 



In the equation of the third degree, however, one other 

 condition, conjoined with that above expressed, will serve to 

 determine infallibly whether all the roots are rational or not. 



To obtain this condition, let us suppose that by making 

 '6oc=zy-\-p we obtain the equation 



y-Q^-R=o. 



Calling the three roots of this new equation a, /3, y (all of 

 which it is evident must be rational if those of the first equa- 

 tion are so), we have 



« + i3 + 7 = 0, 

 Q=:-(a/3 + ay + /3y) = a2 + a^ + /32, 

 R= «|S7. 



From the last two equations it is easily seen that if ^ be any 

 prime factor common to Q and R, Jc^ will be contained in Q, 

 and P in R ; or, in other words, k will be a common measure 



of«j/) 7. 



We have therefore a second condition, that 9 q — Sp^ shall be 

 a negative quantity, which is either prime to 2p^~ 9 5'p+ 27 /', 

 or else so related to it, that the greatest common measure of 

 the cube of the first and the square of the second is a perfect 

 sixth power. 



I now proceed to show the converse, that if these two con- 

 ditions be both satisfied (and it will appear in the course of 

 the inquiry that the first does not involve the second), the 

 roots cannot help being all rational. 



It is evident that the two conditions in question are tanta- 

 mount to supposing that the roots of the proposed equation 

 are linearly connected with those of another s^ — Q^ — R = 

 (by virtue of the assumption ^x=kz-\-p)i where Q may be 

 considered as prime to R; and where 4Q^— 27 R^ is a per- 

 fect square. 



Let now 4 Q«-27 R^ = DS then D2 + 27 R^ = 4 Q^, or 

 D2 + 3 (3 11)2=4 Q3. 



Here, as Q is prime to R, D can have no common mea- 

 sure but 3, with 3 R. 



