RESOLUTION OF ALGEBRAICAL EQUATIONS. 35 
D; for, should A, ye D, Be u,, where D, involves only such 
surds, exclusive of Y, as occur in f (p), this would give us, 
eg ON) ee ne ee) 
Now D cannot be zero, else A, Y would vanish ; but A, is (by hypo- 
thesis) not zero; and the equation, Y = 0, is impossible by 
Def. 9. Hence, since D is not zero, equation (5) is (Cor. 1, Def. 
9) inadmissible. Therefore we cannot have A, Y =D, B,U . 
Consequently, as we established the third of equations (4), we can 
establish similar equations for all the terms in (1), after the first : 
AY SS De BAD, 
May De BLU 
and so on; the terms, A, Y, A, Y, yaaa gees &e., being all different 
from one another, on the one hand; and the terms, B,, U , B, One 
B, ue &c¢., being all different from one another, on the other hand. 
Hence equation (3) becomes, 
c met n af 
(A—B)Y (—D )A,+¥ A—D, )A,+ &. =0; 
where (Cor. 1, Def. 9) the coefficients, A—B, A, Gasp &e., van- 
ish separately. That is, the terms in the series (1), taken in some 
order, are equal to those in the series (2), each to each; A being 
equal to B. 
Proposition III. 
Let f (p) be an algebraical function of a variable p, in a simple 
form ; and let Y,, Y,, &c., certain surds, with the common index > no 
one of them a subordinate of any of the others, be such that all 
their subordinates occur in f (p). Suppose that 
Al A2Q Xa 
Ye = PY, Yo oecees NV 5 p) 
or, as the equation may be written, 
SY RD. Soe dees laas ils te teen (1) 
where Y is merely a symbol used (for the sake of simplicity) to de- 
>, __Ag 
note the continued product of the expressions Y,, SMe oS AY Se ae. 
