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85)
by Q'q' + R'r' + S's' + T't', &c. it is then evident, that this quantity Q'[[underlined]]q'[[/underlined]] + R'[[underlined]]r'[[/underlined]] + S's'+ T't', &c. will express so much of the whole required fluent, as is comprehended between the ordinates EL and FM, or as answers to an increase of EF in the value of [[underlined]] x [[/underlined]]. And thus, if [[underlined]] b [[/underlined]] and B be conceived to be wrote for [[underlined]] x [[/underlined]] and [[underlined]] y [[/underlined]], [[underlined]] [[e or l?]] [[/underlined]] for [[dot over x]] x, and [[underlined]] w [[/underlined]] for [[underlined]] y [[dot over y]][[/underlined]], and the quantity resulting be denoted by Q"[[underlined]] q" [[/underlined]] + R"[[underlined]] r" [[/underlined]] + S"s" + T"t", &c. this quantity will, in like manner, express the part of the required fluent corresponding to the interval FG. Whence that part answering to the interval EG will consequently be equal to Q' [[underlined]] q' [[/underlined]] + R' [[underlined]] r' [[/underlined]] &c. + Q"[[underlined]] q" [[/underlined]] + R"[[underlined]] r" [[/underlined]] &c. But it is manifest, that the whole required fluent cannot be a [[underlined]] maximum [[/underlined]] or [[underlined]] minimum [[/underlined]], unless this part, supposing the bounding ordinates EL, GN to remain the same, is also a [[underlined]] maximum [[/underlined]] or [[underlined]] minimum [[/underlined]]. Hence, in order to determine the fluxion of this expression (Q' [[underlined]] q' [[/underlined]] + R' [[underlined]] r' [[/underlined]] &c. Q"[[underlined]] q" [[/underlined]] + R"r" &c.) which must, of consequence, be equal to nothing, let the fluxion of Q' and [[underlined]] q' [[/underlined]] (taking [[underlined]] alpha [[/underlined]] and [[underlined]] u [[/underlined]] as variable) be denoted by Q-bar alpha-dot and q-bar mu-dot; also let R-bar alpha-dot and [[underlined]] r-bar [[/underlined]] mu-dot denote the respective fluxions of R' and [[underlined]] r' [[/underlined]]; and let, in like manner, the fluxions of Q",[[underlined]] q" [[/underlined]], R", [[underlined]] r" [[/underlined]], &c. be represented by Q-double bar beta-dot, [[underlined]] q-double bar [[/underlined]] Greek eta with dot above, R-double bar Greek beta with dot above, r-double bar Greek eta with dot above, &c. respectively. Then, by the common rule for find the fluxion of a rectangle, the fluxion of our whole expression (Q' [[underlined]] q' [[/underlined]] + R' [[underlined]] r' [[/underlined]] &c. + Q"[[underlined]] q" [[/underlined]] + R"r" &c. will be given equal to Q'q-bar [[u with dot above]] + [[underlined]] q' [[/underlined]]Q-bar [[Greek alpha with dot above]] + R' [[underlined]] [[r with bar above]][[/underlined]] [[u with dot above]] + r'[[R with bar above]] [[Greek alpha with dot above]] etc. + Q"[[underlined]] [[q with double bar above]] [[/underlined]] [[Greek eta with dot above]] + q" [[Q with double bar above]] [[Greek beta with dot above]] + R" [[underlined]] [[r with double bar above]] [[/underlined]] [[Greek eta with dot above]] + r" [[R with double bar above]] [[Greek beta with dot above]] &c. = 0.
   But [[underlined]] u [[/underlined]] + [[underlined]] [[Greek eta]] [[/underlined]] being = GN-EL, and [[Greek beta]]-[[underlined]] a [[/underlined]] = [[GN-EL divided by 2]] (a constant quantity), we therefore have [[underlined]] [[Greek eta with a dot above]] [[/underlined]] = [[underlined]] [[-u with a dot above]] [[/underlined]], and [[Greek beta with a dot above]] = [[Greek alpha with a dot above]]: also u being (=2rp')2alpha-2EL, thence will u-dot=2alpha-dot: which values being substituted above, our equation, after the whole is divided by alpha-dot, will become
  2Q'[[underlined]]q-bar[[/underlined]] + [[underlined]]q'[[/underlined]]Q-bar + 2R'r-bar + r'[[underlined]]R-bar[[/underlined]], &c. - 2Q"[[underlined]]q-double bar[[/underlined]] + [[underlined]]q"[[/underlined]]Q-double bar - 2R"[[underlined]]r-double bar[[/underlined]] + r"R-double bar, &c.=0; or Q"[[underlined]]q-double bar[[/underlined]] - Q'[[underlined]]q-bar[[/underlined]] + R"[[underlined]]r-double bar[[/underlined]] - R'r-bar etc.= q'Q-bar + q"Q-double bar + r'R-bar +r"R-double bar divided by 2, &c.
   But Q"q-double bar - Q'q-bar, the excess of Q"q-double bar above Q'q-bar, is the increment or fluxion (answering to the increment, or fluxion, x-dot) arising by substituting [[underlined]]b[[/underlined]] for [[underlined]]a[[/underlined]], beta for alpha, and [[w]] for [[u]]. Moreover, with regard to the quantities on the other side of the equation, it is plain, seeing the difference of q'Q-bar and q-double bar Q-double bar is indefinitely little in comparison of their sum, that q'Q-bar may be substituted in the room of q'Q-bar + q"Q-double bar divided by 2, &c. which being done, our equation will stand thus:
   [[underlined]]Flux[[/underlined]]. Q'q-bar + R'[[underlined]]r-bar[[/underlined]] &c. = q'Q-bar + r'R-bar &c.
But q'Q-bar + r'R-bar &c. represents (by the preceding notation) the fluxion q'Q' + r'R'&c. (or of Qq + R[[underlined]]r[[/underlined]] &c.) arising by substuting alpha for [[Y]], making alpha alone variable, and casting off alpha-dot. If, therefore, that fluxion be denoted by [[v-dot]], we shall have [[underlined]]flux[[/underlined]].

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Q'q-bar

Transcription Notes:
Many math notations on this page. I used "bar" and "dot" where variables have them marked over the letter, i.e. Q-dot, R-double bar, also used the Greek letter names, alpha, beta, etc. where those are shown. It would be nice to have standard notations listed. Equations and variable notation proved a bit difficult. Apologies to whoever normalizes it to match, though if I get the opportunity to find out how others are managing those things, I will do so myself. Review(II): I posit &c stands in for etc. and have consequently changed any corrected or altered entries back to the original &c. This would reflect the actual meaning of "et cetera" in Latin, where "et" translates to "and"; thus, an ampersand would actually be a logical shorthand. Review: I have changed where this person has typed &c. to Vc.