YouxG — Product of the Squares of the Differences, etc. 753 



It is plain, from inspection, that the sum of the squares of tlie 

 expressions (2) cannot inyolye x-^ ; therefore, this sum must be the 

 same as it would be if Xi were zero ; that is to say, the sum is 



W -p)' + (■/ - p]' ^ ^ {'J -py = - ^p •'• ^ = ^p- 



Again : the sum of the products, two and two, of the squares of the 

 expressions (2), cannot involve x^, seeing that the middle term 

 (the term involving V), in the square of the first expressions (2), 

 is the same, with opposite sign, as the middle term (the term in- 

 volving V), in the square of the second of the expressions (2). 

 Hence, if each of these squares be multiplied by the square of 

 the third of the expressions (2), and the two products be added toge- 

 ther, the terms involving V vrill disappear ; and the result will involve 

 only even powers of a\. And it has been already shown that the pro- 

 duct of the squares of the first and second of the expressions (2) 

 is (Sj^i" -\- pif ; which, in like manner, contains only eveii powers of x^. 

 Therefore, the sum of the three products must be the same as it would 

 be if Xi were zero ; that is to say, the sum of the products is 



and it has been already proved that the product of all the squares is 

 - (27^2 j_ 4^^3-)^ .f,^ 27 q- + 4/;3_ 



Consequently, the equation of the squares of the differences is 



s^ + 6^s^ + 92rz -f 4j?^ 4- 27 q- = 0, 

 the equation which Lagrange has anived at in a very different manner. 



\Note added in Press. — The following somewhat remarkable truth is 

 an immediate inference from Article (12) ia my last Paper, namely : — 



In a cubic equation of which the roots are real, although each 

 root of the derived quadratic always lies between two roots of the 

 cubic, yet it is impossible that either of the two roots of the qua- 

 dratic can ever lie midway between the two neighbouring roots of the 

 cubic] 



