147 
For the very same reason 
( 1 X /li - 2 X Ai -h 1 X Ai - 3 X w -|-2X Ai -3X A j)-h(_ 1 X Ai -_2X A j4-_iX Ai '_3X^ r |-_ 2 X^-_3X fti ) 
and 
( i X^* 2 X Ai + iX. 7li ‘ 3 X h i -f oK h i - 3 X hi ) (_iX w *_ 2 X/ hi + _iX 7u -*_ 3 X A j + _ 2 X Ai •_ 3 X 7ii ) 
are rational functions of the coefficients of G=0, and 
iX 7ti ^X^-J-iX^-gX^ -f- 2 X hi - 3 X 7 , 
is the root of such another given quadratic. Also 
iX 7i 7 - 3 X 7 , -f - — iX 7i j‘_jX 7u -‘_ 2 X 7l! - , and 
iX A7 ** 3X,, *_ 2 X ft i : ^X M 
are rational in the coefficients of G=rO, for the same reason; and 
iXm-Ak-A, 
is the root of such another given quadratic, 
We can now form the cubic whose roots are jX^, 2 X hi and 3 X hi , 
an equation having for coefficients given irrational functions of the 
coefficients of G=0, in which every radical has a determined sign. 
Let (iX hi ) , ( 2 X 7u ) ( 3 X 7n ) be the algebraic roots of this cubic. 
By forming and solving this cubic for the values h— 1, h— 2, 
h— 3, h— 4, we get, and we are content to get, from each of the 
four cubics, its one algebraic root (jX w ), in which there is no am- 
biguity in the radicals. That is, we obtain 
( 1 X,)= 1 J i + 1 K,# 7 + 1 M i 
(iX 2i )= 1 J 7 2 + 1 K £ 2 + 1 Lr+ 1 M 7 2 
( 1 X 3j )= 1 J/+ 1 K/+ 1 L i 3 + 1 M, s 
(iX 4 0= 1 Jf+ I KH ri Lf+ l M!, 
where the left members are all given irrational functions of the 
coefficients of G=0, in which every radical has a determined sign. 
From these equations given for the five values (1, 2, 3, 4, 5) of 
i, we form easily for every such value the quartic whose roots are 
X J i, x K 7 , 1 L i j and jM*. This quartic will have for coefficients given 
irrational functions of the coefficients of G=0, every radical having 
a determinate sign. 
Forming now and solving these quartics for the first five values 
of i, we obtain from each for the one root an irrational function 
( x Ji) of the given coefficients of the sextic G=0, in which every 
radical has a determined sign. That is, we obtain 
