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equation for determining the relation of x and y\ 
when any one of the faid five quantities {viz. abfcifle, 
curve-line, area, fuperficies, or folid) is a Maxi- 
mum or Minimum, and all, or any number of the 
others, at the fame time, equal to given quantities; 
wherein the coefficients g, and h, may be either 
pofitive or negative, or nothing, as the cafe propofed 
may require. Thus, for example, if the length of 
the curve, only, be given, and the area correfpond- 
ing is required, to be a Maximum, our equation will 
e X 
or 2 a X — anfwering to a circle; which 
figurej therefore, of all others, contains the greateft 
area, under equal bounds. 
If together with the ordinate (which, here, is always 
fuppofed given) the abfeiffa, at the end of the fluent, 
be given likewife, and the fuperficies generated by 
the revolution of the curve about its axis be a Mi- 
nimum; then, from the fame equation, we have 
g y X 
1 -h — = 0 : whence 
X x-\- y y 
X iS 
