of any degree however elevated. 459 



shall have 



■ty-x = (x + a$x + »?■&*■ xf 



and therefore 



1 , 1 , 



2 1 l /— 5 





2 2 



fo> £i> £2 °f course do not explicitly involve a. In effect, 

 £ = tf 3 + (^)3 + (02^)3 + 6* &r0 2 ^, 

 f, = 3{# 2 &r-f- {6x)Wx + {Pxfx}, 

 % 2 =3{x' 2 ff l x+(0x)*x+ {e^xfOx}. 

 Now as the expression just obtained for -tyx must be capable 

 of being transformed into M + 0,z + 0# 2 -f . . + O^ -1 , we may 

 at once perceive that the proposed cubic equation will necessarily 

 be such, that the non-symmetric function of its roots, which is 

 represented by £j — £ 2 , shall not involve x. 

 Accordingly we must have 



f^a^b^+b^ 2 , 



?2= a 2 + D i ir + b2* 2 ; 

 a u ag, bj, b 2 being independent of x*. 



Hence I conclude that the equation <£#=0 will, when fM = 3, 

 be subject to the condition 



fi-f2= a i-a 2 (£) 



And a similar result might be obtained for any value of /x 

 greater than 3. 



§4. 



Legendre has indeed been led, by some remarkable researches 

 on the class of equations we have been considering, to infer that 

 the roots of the general equation of the third degree, x, x 1 , x", 

 may be deduced from the successive equations 



,_ a + bx ^ji_ a + ox ' w^ a + bx" 



*~\ + c~x' X \ + dd' °° ~T+c~x T '' 



x l " being equal to the primitive root x. (See his Theorie des 



* It might be seen from other considerations whether b, and b 3 will both 

 of them vanish. But for the purpose in the text no question arises respect- 

 ing their evanescence. 



