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105)
and, consequently, that the sum of as many Q's will, at the same time, be a Maximum or Minimum, because y [[dot above]]is every-where the same, or a constant quantity.

Hence, if the construction of the preceding Lemma be retained (supposing all the Q's, R's, S's &c. to be here expressed as before, in terms of AC, cD', and D'd, &c.) it is plain that the sum of all the Q's, (or of the y[[dot above]]Q's), depending on the said particular triangles (and consequently of all the  y [[dot above]]Q's in general), will be a Maximum or Minimum, when the general relation of y, y [[dot above]], x[[dot above]],(or of AC, cD', D'd,) is expressed by the same equation q + er + fs + gt = 0, there determined: in which q,r,s,t, represent the fluxions of Q, R, S, T, divided by that of x [[dot above]] (= alpha = D'd), and wherein the coefficients e,f,g, will be constant quantities; because it is proved that their values depend intirely on the triangles fG'g, hI'i, kL'l, which remain the same, let the perpendicular (or ordinate) Cc be taken at what distance you will from the given point A; that is, let y stand for which you will of the distances AB, AC, AD, &c. [[2.E.T.?]]

[[underlined]] Corollary. [[/underlined]]

If the sides of the polygon bcdefgh,&c. be diminished, and their number increased [[underlined]]in infinitum[[/underlined]], the sum of all the y [[dot above]]Q's will (it is well known) be expressed by the fluent of y[[dot above]]Q; the sum of all the y [[dot above]]R's, by the fluent of y [[dot above]]R, &c. whence it follows, that, to have the fluent of y [[dot above]]Q (answering to a given value of y) a Maximum, or a Minimum, and the fluents of y [[dot above]]R, y [[dot above]]S, &c. at the same time, given quantities, the relation of y, y[[dot above]], and x [[dot above]], must be defined by the equation q + er + fs + gt = 0, above exhibited; q,r,s,&c. being the respective fluxions of Q,R,S, &c. divided by that of x [[dot above]], (or [[x?]]); this quantity x [[dot above]] or [[x?]], (in finding the said fluxions) being, alone, considered as variable. Hence we have the following

                            GENERAL RULE. 

For the resolution of Isoperimetrical problems, of all orders, take the fluxions of all the given expressions (as well that respecting the Maximum, or Minimum, as of the others whose fluents are to be given quantities), making that quantity (x [[dot above]]) alone variable, whose fluent (x) enters not into the said expressions; and, having divided every–where by the second fluxion ([[x?]]), let the quantities hence arising, joined to general coefficients, [[?]],e,f,g, &c. (whose values will depend on the values given, and may be either positive or negative), be united into one sum, and the whole be made equal to nothing; from which equation the true relation x [[dot above]] and y [[dot above]], and of x and y, will be given, let the number of restrictions be what it will.   

For an example of the general [[strikethrough]] [[rule?]] [[/strikethrough]] Rule here laid down, let the fluxions given be  [[equation: y x [[dot above]][[cubed]]divided by yy[[dot above both y's]],  and x [[dot above]]; the fluent of the former, corresponding to any given value of y, being to be a Minimum, and that of the latter, at the same time, equal to a given quantity. Here, taking the fluxions of both expressions (making [[x?]], alone, variable), and dividing by [[x?]], the quantities resulting will be [[equation: 3y[[xx?]]/[[yy?]] ]] and [[l?]]; so that, in this case, we

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Transcription Notes:
The author uses special symbols for variables. I inserted [[?]] for each special symbol.