A treatise on isoperimetrical problems, and the calculus of variations - Tapa blanda

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1810 Excerpt: ...solution, At dP N--j--= 0. dx By a process similar, but longeron account of the introduction of the quantity ds, Euler deduces from dV = Mdx + Ndy + Pdp + Lds this formula of solution, mT dP Ldy N--j--+-j-2 = 0. dx ds The two preceding formulae have been deduced for the first class of problems that involve one property only. But Euler's method, although more tedious, is not essentially different for problems of the second class. These require the variation of three elements of the curve; and accordingly, we must compute the variation in (F+F'+V")dx, when the three elements, instead of ab, be, ce, become ag, gi, ie; that is, as before we must note the variations in M.dx + N.dy + P.dp + &c. M'.dx+N'.dy'+ P'.dp'+ &c. M".dx + N".df+P". dp" + &c. arising from the translation of the points b and c, to g and i, and from the introduction of two arbitrary quantities or variations such as bg, ci, and thence will result an equation of the form A.bg--B.ci = o, which compared with the equation, R.bg-(R + dR).ci = O, which Euler. had previously obtained, would give-jt-=-----3--, whence R is known. In the formula dV--Mdx + Ndy + Pdp (forming the equation A.bg--B.ci, according to the precepts just given,) A = N'.dx-dP B = N'.dx-dF consequently,-=, ip ancR = N'dx-dP, or =dx(N-£). This is an equation derived from one property y"Vdx: deduce a similar equation from the other fWdx; for d% dx' the curve will be instance, such as f = dx (v--7-), and the equation to + ae = 0, or (-3i)+--) = 0;,,. d(P + a) "dP and since this form is like that of N---j--, instead of two operations for finding the values of R, R', ivovaJVdx-=r a maximum, wnA f JVdx = a constant quantity, we maysubstitute one, and deduce the resulting equation from f...