RESOLUTION OF ALGEBRAICAL EQUATIONS, 21 
1 
speak of the index of the surd so formed, the fraction; is meant. 
ot ea 
For instance, if we regard (c + p) "as generated by the extraction of 
aL 
the fifth root of (c+p) itis a surd of the second order, with the 
index +. From another point of view, it might be described as a surd 
of the second order, with the index 1. 
Def. 4. In the case of a surd of a certain order, we may distin- 
guish the principal surd from its subordinates. Thus, under the 
1 
principal surd, (c+ ./p)*, is involved the subordinate //p. Under 
the principal surd, 
ettien'y b sears 
e being a constant quantity, are involved the subordinates, 
tole 
te) G+), es (l+p ) oH : 
5)a Ep ee 
2.7 4 Dna 8 
re + p) } ; ee (itp ) ; 
the first appearing in the principal surd only in its fifth power; and 
the second only in its second power. A surd which is a subordinate 
of the surd Y, but is not a subordinate of any surd which is itself 
subordinate to Y, may be termed a chief subordinate of Y ; while 
those surds which are subordinates of surds subordinate to Y may 
be called secondary subordinates of Y. 
Def. 5. An integral function of a variable is one in which no surd, 
principal or subordinate, occurs as the denominator, or a term in the 
denominator, of a fraction. For instance, c being constant, and p 
variable, the first of the iar 
Vp é 1 
ae +——__ (Ha Ea 
Je Sle JP ae Jp’ 
is an integral function of p; but the two last are not. 
Cor. A given algebraical function f(p) of a variable p always 
admits of being exhibited as an integral function. For, reduce the 
function to the form *; where each of the quantities N and D is the 
sum of a rational expression, which may be zero, and of a finite 
series of terms, each of them the product of a rational coefficient 
