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106
we have [[numerator]] 3yẋẋ [[\numerator]] /[[denominator]] yẏ [[\denominator]] + e= 0, and therefore ẋ = a [[superscript]] 1/2 [[\superscript]] y [[superscript]] -1/2 [[\superscript]] ẏ (supposing a = -1/3 e). From whence, by taking the fluents, x = 2a[[superscript]] 1/2 [[\superscript] y[[superscript]] 1/2 [[\superscript], or x[[superscript]] 2 [[\superscript] = 4ay, an equation answering to the common parabola.
     If the abscisse of a curve be denoted by x, and the ordinate by y, it is known, that the several fluxions of the abscisse, curve-line, area, superficies of the generated solid, and of the solid itself, will be represented by ẋ, [[square root]] xẋ+yẏ [[\square root]], yẋ, 2py [[square root]] xẋ+yẏ [[\square root]], and py [[superscript]] 2 [[\superscript]] ẋ respectively: if, therefore, the fluxions of these different expressions be taken as before (making ẋ, alone, variable), we shall get 1 + [[numerator]] eẋ [[\numerator]] / [[denominator]] [[square root]] ẋx+ẏẏ [[\square root]][[\denominator]] + fy + [[numerator]] gyẋ [[\numerator]] / [[denominator]] [[square root]] ẋẋ+yẏ [[\square root]][[\denominator]] + hy [[superscript]] 2 [[\superscript]] = 0, being a general equation for determining the relation of x and y, when any one of the said five quantities ( [[underline]] viz [[\underline]] abscisse, curve-line, area, superficies, or solid) is a Maximum or Minimum, and all, or any number of the others, at the same time, equal to given quantities; wherein the coefficients e, f, g, and h, may be either positive or negative, or nothing, as the case proposed may required. Thus, for example, if the length of the curve, only, be given, and the area corresponding is required to be a Maximum, our equation will become [[numerator]] eẋ [[\numerator]] / [[denominator]] [[square root]] ẋẋ+yẏ [[\square root]][[\denominator]] + fy = 0, or a [[superscript]] 2 [[\superscript]] ẋ[[superscript]] 2 [[\superscript]] = y [[superscript]] 2 [[\superscript]] X [[multiplication]] [[bar above]] ẋẋ+ẏẏ [[\ bar above]] (by making a = -e/f); whence ẋ = [[numerator]] yẏ [[\numerator]] / [[denominator]] [[square root]] aa-yy [[\square root]][[\denominator]] , and consequently x = a - [[square root]] aa-yy[[\square root]], or 2ax - x [[superscript]] 2 [[\superscript]] = y [[superscript]] 2 [[\superscript]], answering to a circle; which figure, therefore, of all others,contains the greatest area, under equal bounds.

     If together with the ordinate (which, here, is always supposed given) the abscissa, at the end of the fluent, be given likewise, and the superficies generated by the revolution of the curve about its axis be a Minimum; then, from the same equation, we have 1 + [[numerator]] gyẋ [[\numerator]] / [[denominator]] [[square root]] ẋẋ+ẏẏ [[\square root]][[\denominator]] = 0: whence (making a=-1/g)ẋ is found = [[numerator]] ȧẏ [[\numerator]] / [[denominator]] [[square root]] yy-aa [[\square root]][[\denominator]]; [[strikethrough]] which equation, being  impossible when y is less than a [[/strikethrough]] and, from thence, x=a x [[multiply]] hyp. log. [[numerator]] y + [[square root]] yy-aa [[\square root]] [[\numerator]] / [[denominator]] a [[\denominator]]; which equation, being impossible when y is less than a, shows that the curve (which is here the Catanaria) cannot possibly meet the axis about which the solid is generated; and, consequently, that the case will not admit of any Minimum, unless the first, or least given value of y exceeds a certain assignable magnitude. 
     When any, or all of the above-specified quantities are given, and the contempory fluent of some other expression as [[root]] ẋẋ+yẏ [[\root]] [[superscript]] n [[\superscript]] x [[multiply]] y[[superscript]] m [[\superscript]] x [[multiply]] ẏ [[superscript]] 1-2n [[\superscript]], is required to be a Maximum, or Minimum; the equation (by taking the fluxion of this last expression, and joining it to the former) will then be [[root]] ẋẋ+ẏy [[\root]] [[superscript]] n-1 [[\superscript]] x [[multiply]] 2nẋy[[superscript]] m [[\superscript]] ẏ [[superscript]] 1-2n [[\superscript]] + d + [[numerator]] eẋ [[\numerator]] / [[denominator]] [[square root]] xẋ+yẏ [[\square root]][[\denominator]] +  

Transcription Notes:
Where I have typed a slash mark indicates a division line in a math problem. This symbol: ^ indicates the following character is raised, as in x-squared, which would have a small "2" raised above the x. Some characters are underneath and square-root sign, and I have written [[end]] to show which characters should be included under the square root sign. I would appreciate instruction if there is a better way to indicate these mathematical equations.