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103)
is, by mathematicians, included under that denomination.
But notwithstanding the usefulness and great extent of this subject, nothing (that I know of) had been done thereon farther than the resolution of certain particular cases (such as finding the line of swiftest descent, and the solid of least resistance), 'till the celebrated mathematician McLaurin, in his treatise of fluxions, gave the investigation of an elegant and very easy method, whereby the principal problems belonging to the first order may be solved.
The paper I have now the honour to lay before the Society contains farther improvements on this subject: as it is by far more general than any thing yet offered, and is drawn up with a view to obviate the difficulties attending the resolution of a very intricate kind of problems, and thereby to open an easy way to some very interesting inquiries in natural philosophy, I cannot doubt of its meeting with a favourable reception.  
[center][[underlined]]Lemma.[[/underlined]] I.[/center]
Fig. 9.  At any given points D, G, I, L, in a right line AL, supposing perpendiculars to be erected; and from any other given points c, f, h, k, at equal distances (cD', fG', hI', kL',) from the said perpendiculars, respectively, conceive right-lines cd, fg, hi, kl, to be drawn, to terminate somewhere in the said perpendiculars; let Q, R, S, T, denote any quantities expressed in terms of AC, cD', and D'd, (independent of Cc) and Q', R', S', T', as many other quantities affected in the very same manner with AF, fG', and G'g; and let Q", R", [[insert]] S", T", and Q'", R'", S'", T'", be quantities, still, expressed in the same manner, in terms of AH, hI', I'i, and AK, kL', L'l, respectively: 'tis proposed to find an equation expressing the relation of the inderterminate perpendiculars D'd, G'g, I'i, L'l, so that the quantity Q + Q' + Q" + Q'" may be a Maximum or Minimum, at the same time that the values of the other quantities R + R' + R" + R'", S + S' + S" + S'", and T + T' + T" + T'", are given, or continue invariable.  
[[Put?]] D'd = alpha, G'g = beta, I'i = gamma, L'l = delta; and let the fluxion of Q (supposing alpha variable) be denoted by q'alpha-dot, that of R, by r'alpha-dot, &c. &c. then, since (by the nature of the proposition) the fluxion of Q+Q'+Q"Q'", as well as those of R+R'+R"+R'", S+S'+S"+S'", &c. must be equal to nothing, we therefore have {
q alpha-dot + q' beta-dot + q" gamma-dot + q"' delta-dot = 0
r alpha-dot + r' beta-dot + r" gamma-dot + r"' delta-dot = 0
s alpha-dot + s' beta-dot + s" gamma-dot + s"' delta-dot = 0
t alpha-dot + t' beta-dot + t" gamma-dot + t"' delta-dot = 0

In order now, to exterminate the fluxions alpha-dot, beta-dot, gamma-dot, delta-dot, let these equations be respectively multiplied by [[?]], e, f, g, (yet unknown), and let all the products thence arising be added together, whence will be had [[line above]]q+er+fs+gt[[/line above]] x alpha-dot +[[line above]]q'+er'+fs'+gt'[[/line above]] x beta-dot +[[line above]]q"+er"+fr"+gt"[[/line above]] x gamma-dot +[[line above]]q"'+er"'+fs"'+gt"'[[/line above]] x delta-dot = 0.
Make now, q'+er'+fs'+gt' = 0
q"+er"+fs"+gt" = 0
q"'+er"'+fs"'+gt"' = 0
From whence (there being as many equations as quantities, e, f, g, to be determined), the values of these quantities will be always given in terms of q', r', s', &c. that is, e, f, g, will always be represented by [[end page]]