RESOLUTION OF ALGEBRAICAL EQUATIONS. 25 
ale 
3-H 
25 2 
a 3 3 
sah) ne) (lbp) <a ewp) |p; 
2 being a definite third root of unity. 
Def. S. Let f(p) be an integral function of a variable »; and 
suppose, that, if Y be any surd whatever, principal or subordinate, 
eccuring in the function in its ce power, and having (see Def. 3) the 
index —, c iy less than s, Also, the form of the function being, 
Ff PH=AFALY, +A,Y,+...... ay. Vi ae 
where the coefficients A, A,, &c., are (see Def. 1) rational, and éach 
of the terms Y,, Y,, é&c., is either some power of an integral surd, 
or the continued product of several such powers, suppose that no two 
of the terms, Y,, Y., &c., are identical. Finally, if V be any surd, 
principal or subordinate, occurring in the function in its ™ power, 
and if the form of V. be, 
Reo BeE Be ys BY Ct ott ae rye, 
where the coefficients, B, B,, &c., are rational, and each of the terms, 
Y,, ¥., &c., is either some power of an integral surd, or the con- 
tinued product of several such powers, the index of the surd V 
being 1, suppose that no two of the quantities, Y,, Y,, &c., are 
identical. When these conditions are satisfied, the function f (p) 
may be described as satisfying the conditions of Def. 8. 
Cor. Any given algebraical function f (p) of a variable p admits of 
being exhibited so as to satisfy the conditions of the Definition. For 
should a surd Y, principal or subordinate, with the index +, occur in 
the function in its ec power, e not being less than s, let ws be the 
greatest multiple of s in ¢ ; the excess of c above ws, (which may be 
c ws k 
zero), being &. Thenwemay replace Y by (Y ) Y ; and, since the 
index of the surd Y is 4, Y may be written out so as to involve 
only the subordinate surds of Y. Thus the violation of the first 
condition of the Definiticn, involved in the term Y, is got quit of. 
For instance, 
is} 3 38 
c B i5i BS 
¥ =(1-+”p) -A+“p) +p (1+~p).. 
Next, should any such quantities as Y,, Y,, &c., (see above), be 
