82 On Maxima and Minima of Functions, fyc. 



MATHEMATICS. 



Art. XI. — On Maxima and Minima of Functions of two 

 variable quantities; by A. M. Fisher, Prof. Math, and 



Nat. Phil, in Yale College. 



[From tlie MSS. of the Connecticut Academy.] 



So.ML of the most interesting as well as difficult cases of 

 Maxima and Minima are those, in which a function of two 

 variable quantities, x and y, is required to become the great- 

 est or least joossible, whilst another function of those quan- 

 tities is supposed constant. Let the former of these 

 functions, for the sake of brevity, be denoted by u and 

 the latter by v. Ti)e ordinary method of solving prob- 

 lems of this description is by reducing the two equations, 



^ da;^--^ dy = and r-7 dj? + -T- dy = 0. But as both x and 



y are in this method made variable, it may be difficult, and 

 in some cases impossible, to make the substitutions neces- 

 sary to reduce the two foregoing equations to one. — In most 

 cases it is the relation which holds between x and y, when 

 u becomes a maximum to a given value of v, that is wanted, 

 rather than the absolute value of either of the quantities a 

 or y, expressed in terms of u. It might seem that if a? or y 

 were made constant, and v were allowed to vary, yet by di- 

 viding u by V, and making the differential of the quotient 

 =0, the required relation between x and y would be obtain- 

 ed. This method would indeed give the value of u when 

 its ratio to v is the greatest or least possible to the given 



value of a? or y ; but as the fraction - will vary not only by 



a change in the ratio of x to y, but in that of m to v when - 



is constant, the result obtained would be false. If, howev- 

 ever u become some simple function of v, or vary as (pv, 



when - is constant, the true relation of x to y may be ob- 

 tained by a method which admits of j: or y being made con- 

 stant. This consists in making, not -, but — a maximum 

 or minimum. If this be done, as — can vary only from the 



