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87)
than the relation above assigned.  Then, because the fluents [[underline]]b[[/underline]]B[[underline]]ẋ[[/underline]], [[underline]]c[[/underline]]C[[underline]]ẋ[[/underline]], [[underline]]d[[/underline]]D[[underline]]ẋ[[/underline]] &c. are given, and the same in both cases, it follows, according to this supposition, that this new relation must give a greater fluent of A[[underline]]ẋ[[/underline]] + [[underline]]b[[/underline]]B[[underline]]ẋ[[/underline]] +  [[underline]]c[[/underline]]C[[underline]]ẋ[[/underline]] + [[underline]]d[[/underline]]D[[underline]]ẋ[[/underline]] &c. (under all possible values of [[underline]]b[[/underline]], [[underline]]c[[/underline]], [[underline]]d[[/underline]], &c.) than the former relation gives:  [[underline]]which is impossible [[/underline]]; because (whatever values are assigned to [[underline]]b[[/underline]], [[underline]]c[[/underline]], [[underline]]d[[/underline]], &c.) [[underline]]that [[/underline]] fluent will, it is demonstrated, be the greatest possible, when the relation of [[underline]] x[[/underline]] and [[underline]]y[[/underline]] is that above determined, by the General Rule.  
     To exemplify, now, by a particular case, the method of operation above pointed out, let there be proposed the fluxionary quantity [[numerator]]x[[superscript]]n[[/superscript]]y[[superscript]]m[[/superscript]]ẏ[[superscript]]p[[/superscript]][[/numerator]][[denominator]]ẋ[[superscript]]p-1[[/superscript]][[/denominator]]; wherein the relation of [[underline]]x[[/underline]] and [[underline]]y[[/underline]] is so required, that the fluent, corresponding to given values of [[underline]]x[[/underline]] and [[underline]]y[[/underline]], shall be a [[underline]] maximum[[/underline]], or [[underline]]minimum[[/underline]]. Here, by taking the fluxion, making [[underline]]ẏ[[/underline]] alone variable ([[underline]] according to the rule[[/underline]]) and dividing by [[underline]]ÿ[[/underline]], we shall have [[numerator]]px[[superscript]]n[[/superscript]]y[[superscript]]m[[/superscript]]ẏ[[superscript]]p-1[[/superscript]][[/numerator]][[denominator]]ẋ[[superscript]]p-1[[/superscript]][[/denominator]]= v.  And, by taking the fluxion a second time, making [[underline]]y[[/underline]] alone variable, and dividing by [[underline]]ẏ[[/underline]], will be had [[numerator]]mx[[superscript]]n[[/superscript]]y[[superscript]]m-1[[/superscript]]ẏ[[superscript]]p[[/superscript]][[/numerator]][[denominator]]ẋ[[superscript]]p-1[[/superscript]][[/denominator]] = [[v dot]].  Now from these equations to exterminate v, let the latter be divided by the former; so shall [[numerator]]mẏ[[/numerator]][[denominator]]py[[/denominator]] = [[numerator]][[v dot]][[/numerator]][[denominator]]v[[/denominator]]; & therefore ay[[superscript]][[numerator]]m[[/numerator]][[denominator]]p[[/denominator]][[/superscript]] = v (a being a constant quantity).  From whence y[[superscript]][[numerator]]m[[/numerator]][[denominator]]p[[/denominator]][[/superscript]]ẏ = [[numerator]]a[[/numerator]][[denominator]]p[[/denominator]][[curved line, root function]][[superscript]][[numerator]]1[[/numerator]][[denominator]]p-1[[/denominator]][[/superscript]] X [[multiplication, as opposed to variable x, which is written in cursive]] ẋx[[superscript]][[numerator]]-n[[/numerator]][[denominator]]p-1[[/denominator]][[/superscript]]; and consequently [[numerator]]p[[/numerator]][[denominator]]m+p[[/denominator]] X [[multiplication]] y[[superscript]][[numerator]]m+p[[/numerator]][[denominator]]p[[/denominator]][[/superscript]] = [[numerator]]a[[/numerator]][[denominator]]p[[/denominator]][[superscript]][[curved line, root function]][[numerator]]1[[/numerator]][[denominator]]p-1[[/denominator]][[/superscript]] X  [[numerator]]p-1[[/numerator]][[denominator]]p-n-1[[/denominator]] X [[multiplication]] x [[superscript]][[numerator]]p-n-1[[/numerator]][[denominator]]p-1[[/denominator]][[/superscript]].
     Let there ^ [[be]] now [[strikethrough]]be[[/strikethrough]] proposed the two fluxions x[[superscript]]n[[/superscript]]y[[superscript]]m[[/superscript]]ẋ and x[[superscript]]p[[/superscript]]y[[superscript]]q[[/superscript]]ẏ, the fluent of the former being required to be a [[underline]]maximum[[/underline]], or [[underline]]minimum[[/underline]], and that of the latter, at the same time, equal to a given quantity.  Then the latter, with the general coefficient [[underline]]b[[/underline]] prefixed, being joined to the former, we shall here have x[[superscript]]n[[/superscript]]y[[superscript]]m[[/superscript]]ẋ + bx[[superscript]]p[[/superscript]]y[[superscript]]q[[/superscript]]ẏ.  From whence, by proceeding as before, bx[[superscript]]p[[/superscript]]y[[superscript]]q[[/superscript]] = v, and mx[[superscript]]n[[/superscript]]y[[superscript]]m-1[[/superscript]]ẋ + qbx[[superscript]]p[[/superscript]]y[[superscript]]q-1[[/superscript]]ẏ = [[v dot]].  From the former of which equations, by taking the fluxions on both sides, will be had pbx[[superscript]]p-1[[/superscript]]y[[superscript]]q[[/superscript]]ẋ + qbx[[superscript]]p[[/superscript]]y[[superscript]]q-1[[/superscript]]ẏ (= [[v dot]]) = mx[[superscript]]n[[/superscript]]y[[superscript]]m-1[[/superscript]]ẋ + qbx[[superscript]]p[[/superscript]]y[[superscript]]q-1[[/superscript]]ẏ. Whence pbx[[superscript]]p-1[[/superscript]]y[[superscript]]q[[/superscript]] = mx[[superscript]]n[[/superscript]]y[[superscript]]m-1[[/superscript]]; and therefore pby[[superscript]]q-m+1[[/superscript]] = mx[[superscript]]n-p+1[[/superscript]]. And in the same manner proper equations, to express the relation of [[underline]]x[[/underline]] and [[underline]]y[[/underline]], may be derived, in any other case, and under any number of limitations.
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LXXIII.  [[underline]]Of the best Form of Geographical Maps[[/underline]].  [[underline]]By the Rev[[superscript]]d[[/superscript]].[[/underline]] Patrick Murdock, [[underline]]M.A.F.R.S.[[/underline]]
[[left margin]]Read Feb. 9, 1758.[[\left margin]] I.  WHEN any portion of the earth's surface is projected on a plane, or transferred to it by whatever method of description, the real dimentions, and very often the figure and position of countries, are 
[[right justified]] much [[\right justified]]

Transcription Notes:
There are several mathematical equations that I don't know how to input in this. Tried a method of description, using numerator and denominator and superscript, but these are not in the instructions. The curved line represents a root function. In this case,(a/p) to the root of 1/(p-1).