This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 edition. Excerpt: ...plane. Prove that the ring surface formed by the circumference of the wire cuts itself in two hyperbolic edges, and that its volume is--(c' ] a'--am where K and E are the complete elliptic integrals of the first and second kinds with modulus a/c. math. Trip. 1886. 39. If the modulus /; and the ...
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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1922 edition. Excerpt: ...plane. Prove that the ring surface formed by the circumference of the wire cuts itself in two hyperbolic edges, and that its volume is--(c' ] a'--am where K and E are the complete elliptic integrals of the first and second kinds with modulus a/c. math. Trip. 1886. 39. If the modulus /; and the amplitude / of the elliptic integral F(f, k) be given by k = cos jr/1 2, cos $ = 2-v/3, then will CHAPTER XXXIr. CALCULUS OF VARIATIONS. (section I.) 1482. To ascertain the greatest or least values of which a given function is susceptible under specific conditions, it has been found necessary in the Differential Calculus to allow it to grow, and then to find the magnitude attained when the rate of growth stops. And methods have been formulated by which this rate of variation can be ascertained and tests constructed for the discrimination of maxima values from minima values and from other stationary values which the method may discover. The functions considered in the Differential Calculus have all been expressed directly or indirectly in terms of a set of one or more independent variables not usually involving signs of integration, and if any dependent variables have occurred in the functions under discussion their connection with the independent ones has always been specified and known. We now have a problem of different nature. We are to consider the maximum or minimum value of a function usually expressed by an integration, in which the integrand contains not only an independent variable or set of independent variables, but also one or more dependent variables and their differential coefficients, for which the relationship between the dependent ones with the independent ones is not specified, but remains to be discovered, in order that a...
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