Augustus Edward Hough Love

“ This equation is in general of the 2nth order, and the 2n arbitrary constants which are contained in the complete /primitive-must be adjusted to satisfy the conditions that y, y′, y′′, . . . y (n−1) have given values at the limits of integration. If the function y is required also to satisfy the condition that another integral of the same form as the above, but containing a function φ instead of F, may have a prescribed value, Euler achieved his purpose by replacing F in the differential equation by F+λφ, and adjusting the constant λ so that the condition may be satisfied. ”

“ The terms at the limits and the integrand of the transformed integral must vanish separately, if the variation of the original integral vanishes. The process of freeing the original integral from the differentials of variations results in a differential equation, or a system of differential equations, for the determination of the form of the required curve, and in special terminal conditions, which serve to determine the constants that enter into the solution of the differential equations. Lagrange's method lent itself readily to applications. ”

“ Within the field of these curves we can construct the transversals of the family; that is to say, there is a finite area of the plane, through any point of which there passes one stationary curve of the field and one curve which cuts all the stationary curves of the field transversely. These curves provide a system of curvilinear co-ordinates, in terms of which the value of ∫Fdx, taken along any curve within the area, can be expressed. The value of the integral is the same for all arcs of stationary curves of the field which are intercepted between any two assigned transversals. ”

“ Let Δs denote the arc QP, and ψ the angle which the tangent at P to the varied curve, in the sense from A to B, makes with the axis of x (fig. 5) . Also let φ be the angle which the tangent at P to the stationary curve AP, in the sense from A′ to P, makes with the axis of x. We evaluate (A′Q) − (A′P) approximately by means of a result which we obtained in connexion with the problem of variable limits. ”

“ The formation of the first variation of an integral by means of a weak variation can be carried out without difficulty in the case of a simple integral involving any number of dependent variables and differential coefficients of arbitrarily high orders, and also in the cases of double and multiple integrals ”

“ Thus the argument fails to distinguish between a minimum and an inferior limit (see FUNCTION) . In order to prove Dirichlet's principle it becomes necessary to devise a proof that, in the case of the integral 1, there cannot be a limit of this kind. ”

“ This problem of the catenary was attempted without success by Galileo Galilei (1638) . (ii) The resistance of a medium to the motion of a body being assumed to be a normal pressure, proportional to the square of the cosine of the angle between the normal to the surface and the direction of motion, it is required to determine the meridian curve of a surface of revolution, about an axis in the direction of motion, so that the resistance shall be the least possible. This problem of the solid of least resistance was solved by Sir Isaac Newton (1687) . ”

“ Legendre's remark amounts .to admitting a variation of Newton's curve, 'which is not a weak variation. Variations which are not weak are such that, while the points of a curve are but slightly displaced, the ta-ngents undergo, large changes of direction. They are distinguished as strong variations. A general theory of strong variations in connexion with the First Problem, and of the conditions which are sufficient to secure that the integral taken along a stationary curve may be an extremum, was given by Weierstrass in lectures. ”

“ It has been proposed to establish the existence of V by means of the argument that, since I cannot be negative, there must Dmch be, among the functions which have the prescribed ers boundary values, some one which gives to I the smallest principle. ”

“ H. Hancock, Lectures on the Calculus of Variations (Cincinnati, 1904) ; and O. Bolza, Lectures on the Calculus of Variations (Chicago, 1904) , give accounts of Weierstrass's theory. Kncser has made various extensions of this theory. Bolza gives an introduction to Hilbert's theories also. The following memoirs and monographs may be mentioned: J. L. Lagrange, “ Essai sur une nouvelle méthode pour determiner les max. et les min. des formules intégrales indéflnics, " Misc. Taur. (1760~62) , t. ii., or Quvres, t. i. (Paris, 1867) ;.A. M. Legendre, “Sur la maniere de distinguer les max. des min. ”