Introduction. 3 



which it was previously fractional, we are said to Integralize the 

 expression with respect to those quantities. Thus the expression 



a'x^ 2ab b'x' 

 b x^ a 

 is integralized with respect to a and b if it is multiplied by ab. 



Similarly, if, by any operation, we render an expression 

 rational with reference to certain quantities, in respect to which 

 it was previously irrational, we are said to Rationalize the expres- 

 sion with respect to the quantities named. Thus if we square 

 the irrational expression 



'^ x'-^V~ab xy^-y-" 

 it is rationalized with respect to x and y. 



6. Definitions. The Degree of a term with respect to any 

 quantity or quantities is the sum of the exponents of the quan- 

 tities named. Thus ab'x'^y is of the third degree with reference to 

 X, of the first degree with reference to y, of the fourth degree with 

 reference to x and y, of the third degree with reference to a and 

 b, etc. But the degree with reference to any quantities is not 

 spoken of unless the term is rational and integral with respect to 

 those quantities. Thus we do not speak of the degree of such a 



\^' 



term ^s — , with respect to either a or x. 



The Degree of a polynomial with respect to any specified quan- 

 tities is the degree of that one of its tenus whose degree (with re- 

 spect to the same quantities) is highest. Thus, x^—alx'^y^-\-cxy 

 is of the third degree with respect to x, of the second degree with 

 respect to y, and of the fourth degree with respect to x and y. 

 But the degree of a polynomial is not spoken of unless the poly- 

 nomial is rational and integral with respect to the quantities 

 specified. 



It can easily be seen that the degree of the product of several 

 polynomials is the sum of their separate degrees. Thus 



(^'+^^+y) {xy^bx-y) 

 is of the fifth degree with respect to x and y ; of what degree is it 

 with respect to jf ? with respect to_y? 



The Degree of an Equation is the degree of the term of highest 

 degree with respect to the unk7iotvn quantities. But both mem- 



