2 Algebra. 



jc and J/ ; that is, is an integral function of a and d, but 2i fractional 

 function of jr and r, the word fractional meaning just the opposite 

 to integral. 



3. Definition. An Expression is Rational with respect to 

 any quantity or quantities, or is a rational function of those quan- 

 tities, when the quantities referred to are not involved in any 

 manner by the extraction of a root. Thus 



X 



is rational with respect to x, but irrational with respect to c and 

 d\ that is, it is a rational function of x, but an irrational function 

 of c and d, the term irrational being used in just the opposite 

 sense from rational. 



4. An expression may be both rational and integral with re- 

 spect to certain quantities, in which case it may be spoken of as a 

 Rational Integral Expression with respect to those quantities, or 

 as a rational integral function of the quantities. In the sam6 

 way we may speak of an expression which is rational and frac- 

 tional with respect to certain quantities as a Rational Fractional 

 Expression with reference to those quantities, or as a rational 

 fractional function of the quantities. In like manner we may use 

 the terms Irrational Integral Expression and Irrational Fractional 

 Expression, or Irrational Integral Function or Irrational Frac- 

 tional Function. 



In the following examples the student is expected to answer 

 the question, What kind of an expression? with reference to the 

 quantities specified opposite each. 



J. ax^-\-a^x''-\-a^x. With respect to Jt* ? to a ? to Jt: and a? 



2. — {\-\-~y \—c\ With respect to ^? to r? 



a 



J. bx'' 7-^xy. With respect to a'? tOJ^'? to x and j? 



v/^ x'-^^ b-^Wc With respect to Jt:? to^y? 



ay^-\-by-\-c ' to x andjj'? to a, b, and r? 



5. Definition. If by any operation we render an expres- 

 sion integral with reference to certain quantities, in respect to 



