On Maxima and Minima of Functions, S^c. 83 



variation of a; in regard to y, df — ) taken in respect to one 



of the variable quantities x and y, and made =0, will give 

 the same relation between x and y as if t> were made con- 

 stant, and X and y were both allowed to vary. 



This result is concisely expressed in the following 



THEOREM. 



If u and V be any functions of x and y, and u become any 

 any simple function of v, (that is, vary as t>" , log a;. , «% 



&c.) when - is supposed constant, either of the equations 





V-— =0, or — ~^=0, gives u a maximum or minimum 

 to a given value of v. 



The utility of this theorem will chiefly appear in the so- 

 lution of Isoperimetrical problems. If x and y be the va- 

 riable quantities in the equation by which the species of any 



geometrical figure* is expressed, and - be supposed con- 

 stant, while the arbitrary constant in the equation is made 

 to vary, the figure will continue similar to itself; and there 

 fore if V and u be either of the quantities compared in iso 

 perimetrical problems, cpv will become v"; and u cc V". If 

 u be the length of a curvilinear figure, or a line drawn in or 

 about it in any given manner, and v the area of the same 

 figure, n = \ : if m be a solids and v be the whole or any 

 constant part of hs superficies, n = | : ii v and u be both 

 solids, superficies, or lines, w=l, &c. 



* Those figures only are intended, which are capable of being defined by 

 one arbitrary constant quantity, in addition to the two variable ones x and 

 y. If the figure be a curve whose absciss and ordinate are x and y, and into 

 the equation of which more arbitrary constant quantities than one neces- 

 sarily enter, as is the case in most curves of the higher orders, different 

 curves may be constructed to the same absciss and ordinate, merely by va- 

 rying the relation of those constant quantities ; so that although - be suppos- 

 ed constant, the curve does not necessarily continue similar to itself. But 

 the straight line, the circle, the parabolas, and the hyperbolas when refer- 

 red to their asymptotes, are included in the first class, together with most oi 

 the other curves, both algebraic and transcendental, which are the most 

 interesting in their properties, and have received particular names. The 

 same thing is true of their superficies and solids of revolution. 



