A treatise on infinitesimal calculus Volume 2 ; containing differential and integral calculus, calculus of variations, applications to algebra and geometry, and analytical mechanics

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. 1854 Excerpt: ...which are primarily necessary to « having a maximum or a minimum value. 208. Although it is desirable, both for symmetry and for the discussion of an expression in its most general form, to retain all the terms in 5m thus far, and although in many of our subsequent examples we shall retain them throughout, yet it is necessary somewhat to abridge them, that we may point out some general properties of the above equations. First, let the difference between (7), (8) and (9), (10) be observed: (7) and (8) involve limiting values of bx, by, and of their differentials; whereas E = 0, and H = 0, being differential expressions, will after integration give general relations between x and y, and therein the required functional connection; and the same function will be deduced both from B = 0 and from H = 0, provided that (and this is a necessary condition) the same limiting values are taken in the integrals of both equations: for the form of the function involved in them will depend on the form of function of 12, and from 12 they are derived by a similar process; and therefore the same functional form will appear in the final result of each. Again, let us suppose that there is no variation of x, save at the limits; and that therefore the shifting of any point from a curve to the next consecutive curve is due to a variation of y only; then bx = 0 (except at the limits), dbx = d2bx =... = 0: so that (6) becomes 209. Suppose that 12, which involves d"'y, is not linear with respect to dmy, then Ym is a function of dmy, and therefore dmYm involves d2my: the equation H = 0 involves therefore (Pmy; and as, during the process of integrating H = 0, an arbitrary constant is manifestly introduced at each successive integration, so does the complete integral involve 2m ...