On Marrma an cl- Minima of Functions , SfC 



MATHEMATICS, 



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Art- XI. — On Maxima and Jllimma of Functions of two 

 tariahh quantities; hy A. M, Fisher, Prof Math, and 

 JYfft. Phil, in Yale College, ^ 



[From the MSS. of the Connecticut Academy.] 



Some 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 possible, whilst another function of those quan- 

 .titie3 is supposed constant, luet the former of these 

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

 the latter by t?. The ordinary method of solving prob- 

 lems of this description fs by reducing the t'i^o equatioiis, 



-r-da?4-Vdy = and r— drH — r- 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 oft?, that is wanted, 

 rather than tlie absolute value of either of the quantities oc 

 ory, expressed \\\ terms of r. It might seem that if <r or y 

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

 viding » by % and making the difTerentlal 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 oc least possible to the given 



value of ^ or y ; but as tbe fraction - will vary not only by 



k change m the ratio of x to y, but in that of m to r when - 



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

 ever u become some siaiple function of t?, or vary as (pt?, 



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

 tamed by a method which admits of j? or y being made con- 

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

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



