RESOLUTION OF ALGEBRAICAL EQUATIONS. 139 
pression L involves only such surds as occur in F, (2) or ty (z); but, 
since U is (by hypothesis) not a subordinate of any surd in the func- 
tion F,(«), the substitution of 2; U for U makes no change on the 
surds appearing in the function: that is, the surds in f (x), and there- 
fore also those in “fo (x), are identical with those in F, (2); and con- 
sequently the surds in L are all found in F,(z). Now the simple 
factors of F(a) are (see Prop. VIII.) the unequal cognate functions 
of f (p) obtained by assigning definite values to those surds in f (p) 
which are also present in F,(%), and taking the cognate functions 
without reference to the surd character of the surds so made definite. 
Therefore (Cor 4, Prop. VI.) no expression such as L can involve 
merely such surds as appear in F,(%). Hence F.(z) cannot but be 
equal to some termin (5). Let F. (7) = “fe (z). This implies that 
the coefficients of like powers of # im these expressions are equal. 
Let bea power of @in F,(@) involving in its coefficient the surd 
: : th 
Y in one of its powers; the coefficients, D and D,, of the E power 
of zin F,(#) and fe (a) respectively being, 
les) 
D=...... +0,Y U + &e., 
in(a=0) qlpp ee wT 
Die Weg ot Barend dt auld (NO IIR) its 
where such terms as 6, are clear of the surds Y and U, and not zero; 
fee 
and no two terms such as that written Y U are identical; % not 
bemg zero. Since s is a prime number, and [F, (2) satisfying the 
conditions of Def. 8] / is less than s, we can find whole numbers, w 
and w,, less than s, and such that 
Dees se Cy aes 
k 
or, if Y be represented by V, 
Y pi (ah a ia) 
Now (Yv vie when expressed as an integral function satisfying the 
conditions of Def. 8, involves only the subordinate surds of Y. 
Therefore, by the equation found, we can eliminate Y from Fe (2), 
introducing in its room powers of V, but no powers of any other surd 
that was not previously in the function. Let F.(«), as thus ex- 
hibited, be written F, (z). A term in F. (a) is 5 V U - Should 7 
