of Algebraic Equations of the Fifth Degree. 551 



From which there will result, on writing i** instead of », 

 a pi =: a^in. 



We might have arrived at the expression for a^ from con- 

 sidering that y« + ^y + a,y^i which expressed as a function 

 of^ and M would become »'*(»'* + » + «,) ^ + » (i+ i* + «Jm, must in 

 vanishing assume the form 0^ + Om. 



It appears then that a,n has only two different values, a, and 

 a^2. Hence all functions symmetric relatively to a^ and a^ 

 will remain unaltered when we substitute any one of the three 

 roots, 1^, <^, «"*, instead of «. 



And in accordance with this we find 



«^ + «,2=-(l +i + »^ + *^+«'*)=-(-l), 



a, and a^ are in fact the roots of the equation 



a^ — a - 1 = ; 

 which solved as a quadratic equation will give 



1+ \/5 



a = 



2 



10. Another consequence of the properties of «/ must here 

 be pointed out. 



Representing any one of the ten equations of the system by 



and observing that a,n is equal to 1— a^2n, we see that 



ya + 2/c + ^/^ 2^b = 2^a + yb + yc - «;2«yb 

 = -(yd + Ve +«i2«?/b); 

 a, b, .. e having, for greater simplicity, been introduced in- 

 stead of the accented indices a, /3\ . . g\ 



Thus it appears that p^ and p^ cannot be determined by 

 means of any two equations expressible by 



ya +yc + «i"yb = 0,| ^^^j 



Vd + ye .+ «,2"^b = o,j 

 which, although seemingly independent of each other, are in 

 reality reducible to a single equation. 



11. In discussing equations of the class (c), the following 

 definition will be found useful : — 



Of the two functions J/a + Vc + «i" Vhi Vd + 2/e + <^/« Vb, 

 which are of the same form, and which taken together include 



