On Maximii,(uii JMinma.of FuMtiojff, S^, 



variation of a; in regard to y, dr-^Vtakeu in respect to one 



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

 the same relation between x and y is if v were made con- 

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

 This result is concisely expressed in the followinj^" 



**rHEOHfc:M. 



. 4 - ^ #* 



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

 any simple function of r, (that is, vary as z?'* , log i?. , a'^i 



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



— ^— =0, or — - — =0, gives n a maxminm or minimum 

 to a given value oi v. ^\^ 



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

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

 riable quantities in the equation by which the specie^ of 4ny 



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



stantj while the arbitrary constant in the equation is made 

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

 fBre if y and u be either of the quantities compared in iso 

 perimetrical problems, (pv will become tf^, and m CC u". If 

 II be the length of a curvilinear figure, or a line drau*n in or 

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

 figure, n = ^ : if w be a solid, and v be the whole or an 

 constant part of hs superficies, n = |: if v and u be bot 

 solids, superficies, or lines, nr=l, &c- 



• Those figures ouly are intended, which arc capable of beings defined by 

 oae arbitrary constant quantity, in addition to the two rariable ones x and 

 y. If the figure be a curve whose absciss and ordiniite are r and y, and into 

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

 sarily enteri as ia the case in most oarres of the higher orders, different 

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



I7^"a the relation of those constant quantities ; so ttiat althou^ - be snppos- 



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

 the straight line, the circle, the parabola^, and the hyperbolas when refer- 

 red to their asymptotes, are incldded in ti^e^first class," tog-ether with most of 

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

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

 ?^Bae thioff is true of their suDerficiai and Mollis of revolution, ' * 



