26 RESOLUTION OF ALGEBRAICAL EQUATIONS. 
identical, the terms containing the identical quantities, as described, 
may be combined into a single term. Jor instance, 
2 3 2 3 
Ff (p)=pt+ {oe ee my +p Pain Set 
3 
=p+(1+p) el oun \ b 
Def. 9. An irrational function, f (p), of a variable p, is said to be 
in a simple form, when no equation such as, 
A+BUSCV5...... Ppye. tt se) HOG ies Wah) 
can subsist; where the coefficients, B, C, ...... , H, all of them dis- 
tinct from zero, are (see Def. 1) rational; A likewise being rational ; 
and each of the terms, U, V, ...... , T, is either some power of an _ 
integral surd occurring in f (7), or the continued product of several 
such powers; the expression on the left hand side of the equation 
satisfying the conditions of Def. 8. Let it be observed, that, in this 
paper, when we speak of a surd occurring in a function, we mean 
that the surd appears in the function, as a principal or subordinate 
surd, in some one or more of its powers, but not necessarily in the 
first power. Thus, the surds which occur in the function, 
2 5 
p+i+(p— pl) +(p— 1), 
1 
are, Vp? — 1, and, (p — “p> — 1) . The first occurs in its first 
power; the second, in its second and fifth powers. This being kept 
in view, we may instance, as violating the condition above men- 
tioned, the function, 
ney Silat jit Oe ok 
CD) 1p ce Vp) (pA Vo Sete oe actrees (2) 
For the equation, of the form (1), subsists : 
a a 
ee SES Sees a 
= epi ior opis lyr (Ne 
Hence f (p), as exhibited in (2), is not in a simple form. 
Cor. 1. The Definition implies, that, should an irrational function 
of a variable p, in a simple form, and equal to zero, present itself in 
the form, 
(p= A+ BU OV Sine iBT, 
where U, V, &c., are terms of the same kind as in equation (1), and 
