2 DEFINITIONS. 



3. The student has probably already had occasion to 

 consider the meaning of the terms "variable quantity" and 

 " function" which we have here introduced. In treatises on 

 the conic sections, for example, the equation y 2 <</ax occurs, 

 where x is a general symbol to which different numerical 

 values may be assigned, and a is a symbol to which we 

 suppose some invariable numerical value assigned, and which 

 is therefore called a "constant." For every value given to x 

 we can deduce the corresponding numerical value of y. In 

 the equation y = 2 *Jax, since the value of y depends upon 

 that of a as well as that of x, we may say that y is a function 

 of a and x. Hence such symbols may be used as F(a, x) 

 to denote a function of both a and x ; and such an equation 

 as y = <f> (x, z, t) indicates that y is a function of the three 

 quantities denoted by the symbols x, z, and t. 



4. In the equation y = 2 Jax, if we know that a is to be 

 a constant quantity throughout any investigation on which 

 we may be engaged, we shall frequently not require to be 

 reminded of this constant, and shall continue to speak of y 



as a function of x. So the equation y = - V( s &*) ma J be 



a 



represented by y =<(#), where we express only that sym- 

 bol x which throughout our investigations will be considered 

 variable. 



5. If the equation connecting the variables x and y be 

 such that y alone occurs on one side, and on the other side 

 some expression involving x and not y, w r e say that y is 

 an explicit function of x. When an equation connecting x 

 and y is not of this form, we say that y is an implicit function 

 of x. Thus if y = ax* + bx + c, we have y an explicit function 

 of x. If ay* 2bxy + cx z + g = we have y an implicit func- 

 tion of x. The words implicit function assume that y really 

 is a function of x in the sense in which we have used the 

 word function. This assumption may be seen to be true in 

 the example given, for we can by the solution of a quadratic 

 equation exhibit y as a function of x; or rather we can infer 

 that y must be one of two explicit functions of x, namely 



b.r + A/f f/> 2 ac) x* nq\ bx \f\(h 2 af) x* oq\ 1TT 



either - * or . We 



a a 



shall return to this point hereafter, in Art. 58. 





