88 On Maxima and Minima of Functions, ^c. 



If we attempt to solve this problem in the usual way, by 



jT 



makine — - — —- —constant and = b, we obtain, after a 



tedious process, two equations between x and y, one of 

 which is a quadratic in regard to y, and the other a cubic in 

 regard both to x and y. This appears to be the simplest 

 form to which the solution, if it may be called such, can be 

 brought, and it is only by an artifice not very obvious, that 

 even this degree of simplicity can be attained. 



The converse of this problem may, however, be readily 

 solved in the usual manner, in consequence of the simplicity 

 of the expression for the solidity. If with a given solidity, 

 the superficies be required to become a minimum, since the 

 solidity varies as x^y, [x being as before, the absciss, and y 



the parameter,) put x-y=b-; then vT=-^, and by substi- 



tution in the expression for the surface, 4byi-\-y^\'y —y^ 



=min., which by putting z in the place of yf , affords the 

 following equation: 3z- — ■jzb + b^=o. This equation is of 

 the same form with those already obtained, and it is evident 

 that the same relation ought to exist between z {=y^) and 

 b, {=xy^), as between y and x. 



But the method adopted above is equally applicable to 

 cases when both the functions u and v are complex, as will 

 appear in the following problems. 



Prob. IX. 



It is required to determine when a parabola of given 

 length will describe the greatest possible superficies by it? 

 revolution about its absciss. 



In this example, v denoting the length and u the super- 

 ficies, (the parameter being made constant and =2a,) 



Ji.= max, or, which is the same thing, vu~'^ = m\n. Sub- 



stituting the normal x (=\/ y^ -\-a^), in place of y in the 



ordinary expressions for the length and superficies, the f or- 



tA-v x^ —a^ 



mer will be found to vary as xv' a? ^ —a* +«* h.l. ; 



