4 Algebra. 



bers of the equation must be rational and integral with respect to 

 the unknown quantities and the indicated operations must be per- 

 formed ; otherwise the degree is not spoken of. 

 What is the degree of 



(jt" -f y ) (jry + 1 ) = 208-ry ? 



Instead of speaking of expressions as being of the first or of the 

 second or of the third degree, they are commonly designated by 

 adjectives borrowed from geometry as linear or quadratic or cubic 

 expressions respectively. An expression of the fourth degree is 

 sometimes called bi-quadratic, meaning twice squared. 



In place of the expression, "of the second degree in respect to 

 .ar," it is common to say, "of the second degree in x/' 



7, DkfinitioNvS. a polynomial is Homogeneous with respect 

 to certain quantities when all its terms are of the same degree 

 with respect to those quantities. Thus a^-\-d'b-\-ab~-\-b^ is homo- 

 geneous with respect to a and b. 



An equation is Homogeneous when all the terms are of the same 

 degree with reference to the unkjiown quantities. Thus the equa- 

 tion xy-\-j/^-\-x^=o is homogeneous, but xy-\-y^-\-x^=2o is not 

 homogeneous. 



It is to be noted ^ here that we use the term homogeneous equa- 

 tion in the strict .sense, following the established use of the term. 

 But in some American text books homogeneous equation includes 

 equations like x-y-{-y^-\-x^=20, that is, no account is taken of 

 terms involving nothing but known quantities. 



8. Definition. An expression is Symmetrical with respect 

 to two quantities if the expression is unaltered when the two 

 quantities are interchanged. Thus x^-{-y^ is symmetrical with re- 

 spect to X and J' ; for putting j/ for x and x for j we oh\.3.h\y^-\-x^^ 

 which is the same as the original. Also x''-\-ax+a'' is symmet- 

 rical with respect to a and x. Is x^-\-2xy—y^ symmetrical with 

 reference to x and y ? 



An equation of two unknown quantities is symmetrical when 

 the interchange of the unknown quantities throughout does not 

 modify the equation. Such is 



X -\- xy -f- xy'^ +y= 1 024. 



