156). RESOLUTION OF ALGEBRAICAL EQUATIONS. 
surd involyed in it of the form (1); nor with any surd involved in it, 
which, while not subordimate to any other in the function, is of the 
form (2). 
Proposition XIII. 
Let f (p) be an integral function of a variable p, in a simple form, 
containing no surd such as Y in (1) Prop. XII, nor any surd, which, 
while not subordinate to any surd in the function, is of the form 
shown in (2) Prop. XII; and having no two surds similarly in- 
volved init. Let Y be a surd inf (p), with the index 4 » not subor- 
dinate to any other in the function; and let the function, arranged so 
as to satisfy the conditions of Def. 8, be, aS 
AiCoi= Ait AgM de AY sh Sica se seeps vceaMibesdeltoalale (13 
where A,;, A,, &c., none of them zero, are clear of the surd Y; 
A also being clear of the surd Y; and Y", Y", &c., are distinct powers 
of Y. Suppose that T and T, are two chief subordinates of Y, with 
the indices + aud + ; but that neither of them is a subordinate of 
1 
any other surd in the function f(p). When T is changed into z,T, 
z, being a o root of unity, distinct from unity, let f(p), Y, A, A., 
&¢., be transformed into f,(p), Y,, B, B,, &c.; and, when T, is 
changed into z,T,, 2 being a of root of unity, distinct from unity, 
let these same expressions become f,(p), ,Y, 6, 6,, &c.; so that 
Gp) = BSB YG, eRe ae oe, } a 
and, f,(p) = b + Dl GP NERIACNG ye &e. 
Then, if f;(p) =f2(p), it can be proved by the same reasoning as 
in Prop. II, that the terms, 
BeYy BL hpeoy 
taken in some order, are equal to the terms, 
NES CORR CO ors 
But should the numbers o and o, not be both equal to s, the equation, 
Bi. You =a Desh Yun Yo benaret decas Gat -aueles a ene (3) 
cannot subsist. 
