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104
by quantities depending on q', r', s', &c. (or on AF, G'g, &c.) exclusive of q, r, s, t, (or of AC and D'd), which have nothing to do in these last equations.
  But, because all the terms of the equation q+er+fs+gtx[[?]]+q'+er'+fs'+gt'xβ[[?]], &c.=0, after the first (q+er+fs+gtx[[?]]) do vanish (by their coefficients being made equal to nothing), it is evident that q+er+fs+gt must also be =0: which is an equation expressing the general relation of AC, cD', and D'd, with regard to the other [[strikethrough ?]] proposed quantities AF, fG', G'g, &c. whereon the coefficients e, f, g, depend: and this relation will, evidently, continue the same, at whatever distances from the line AI., the points c, f, h, k, are taken, as these distances have nothing to do in the consideration, all the propos'd quantities (as well the Q's as R's, &c) being (by hypothesis) express'd in terms intirely independent thereof.

         [[underline]] Lemma[[underline]] II.
Fig. 10. Upon a given right-line BI, suppose perpendiculars Bb, Cc, Dd, &c. to be erected at equal distances; and upon the same line BI, as a base, suppose a polygon BbcdefghiklI. to be constituted, having its angular points b, c, d, &c. posited in the said perpendiculars., let y denote the distance of any of those perpendiculars (Cc, Dd, &c.) from any given point A, in IB produced; and, supposed bC', cD', dE', &c. to be drawn parallel to AB, let the base of any of the little triangles bC'c, cD'd, &c. be represented by ẏ, and the perpendicular corresponding by ẋ (ẏ being given, or the same, in every triangle, and ẋ indeterminate): then, supposing Q, R, S, T, to denote any quantities express'd in terms of y, ẏ, and ẋ, it is proposed to find an equation exhibiting the general relation of the quantities y, ẏ and ẋ, so that the sum of all the ẏQ's (resulting from the several triangles) may be a Maximum or Minimum, at the same time that the sums of all the ẏR's, ẏS's, &c. are given quantities.
   Because the values of the quantities ẏQ, ẏR, ẏS, ẏT, depending on the different triangles bC'c, cD'd, &c. are supposed to be no-ways affected by the distances (Bb, Cc, &c. of the bases of those trangles, from the base BI of the polygon, it is evident, that those values may be changed, by altering the species of one, or more, of the said triangles at pleasure, without any-ways affecting the values depending on the other triangles: for another polygon IB12345,&c. may be so described as to have all its sides, respectively, parallel to those of the former, excepting only those (23, 56, 78, 910) you would have to be different: so that the whole variation in the several sums (whether of the ẏQ's, ẏR's, or ẏS's, &c.) will depend intirely upon the difference of the particular triangles 2q3, cD'd, 5t6, fG'g, &c. assigned.
   Since, therefore, the values of they ẏQ's ẏR's, ẏS's, &c. may be varied, at pleasure, by altering the species of any number of corresponding triangles (2q3, cD'd; 5t6, fG'd; 7w8, hI'i; gy10, kI'l), while the other triangles, and the values depending on them, remain the same, it is manifest, that, when the sum of the ẏQ's, answering to all the triangles, is a Maximum or Minimum, the sum of any number of them, taken at pleasure (other things remaining the same), will likewise be a Maximum or Minimum and.
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Transcription Notes:
Not sure how to transcribe the symbol that looks like an incomplete infinity used in many of the equations. Nor the dot next to the beta. How to indicate the vertical line over y and x in the Lemma II section? Don't want it to get confused with y' and x'.