RESOLUTION OF ALGEBRAICAL EQUATIONS. 157 
For suppose, if possible, that equation (3) subsists. Then the 
reasoning of Prop II. makes it plain that B,Y, is equal to 6, (,Y ), 
and soon. We will therefore assume this. The s‘* powers of A,Y 
B, Y,, and 4,(,Y ), arranged so as to satisfy the conditions of Def. 8, 
are of the forms, 
es m n M N 
(A.Y) =D+D,T T, +D.T T, + &c., 
ec s m m n M M N 
GEV Dia), DT 2 De TD tea) +... (4) 
e 8 n m n N M N 
S60Y)} =D+2D,T T, +2, D,T T, +&e., 
where D, D,, &c., are clear of the snrds T and T,; no two terms in 
the series, T” T, , TT , &c. being identical with one another. There 
must be at least one term in the series D,, D3, &c.; else (AY yi would 
be reduced to D; in which case (Cor. Prop. X.) the surd Y would 
be of the inadmissible form given in (1) Prop. XII. But, from (3) 
and (4), 
m n N 
fk Dias) at Ty Diz, 2.) we. = 0. 
Hence, by Cor. 1. Def. 9, the coefficients of T T., T qT. 5 OCC. 
vanish separately. But, since the expressions on the right hand side 
of (4) satisfy the conditious of Def. 8, the terms D,, D,, &c., do 
_ not vanish. Therefore 
and so on: from which it follows that o, =o; and also that zg=2? ; 
where u is a whole number, less than o, and such that 
Un = Wyo +m, (5) 
uN=w,c + M, 
and so on; ~, w,, &c., being whole numbers. Let V be put for 
min 
T T,. Since o, has been proved equal to o, the surds T and T, have 
a common index; and V may be considered a surd with the same 
index as that of T and T;. Take W and w,, whole numbers less than 
o, and such that 
Wm = w+ M. 
VoL. V. N 
