A History of the Progress of the Calculus of Variations During the Nineteenth Century

Macmillan and Company, 1861 - 532 páginas

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Página 140 - It is true that one of the fundamental principles of this method consists in removing as much as possible the differential coefficients of the variations which occur under the integral sign; but the calculus of variations only indicates this operation and refers the execution of it to the Integral Calculus. CHAPTER VI. DELAUNAY. 133. THE Academy of Sciences at Paris proposed the following as the subject of competition for their great mathematical prize in 1842 ; To find the limiting equations which...
Página 229 - Jacobi in 1837, and the memoir which Jacobi then published has given rise to an extensive series of commentaries and developments. Before however we proceed to Jacobi's investigations, we will give an analysis of Legendre's memoir and of some others connected with it. • 197. Legendre's memoir is entitled Memoire sur la maniere de distinguer les maxima des minima dans le Calcul des Variations.
Página 539 - One of the most charming works on Natural History . . . written in such a style, and adorned with such a variety of illustration, that we question whether the most unconcerned reader can peruse it without deriving both pleasure and profit.
Página 142 - ... can be calculated. It will be necessary to effect successive integrations, and to take each integral between appropriate limits, and these can be determined in the following manner. The order of the successive integrations being arbitrary, we can suppose that we integrate first with respect to z, then with respect to y, and then with respect to x. In the first integration y and x are regarded as constants, and the integration with respect to z extends over all the values of z which render f(x,...
Página 543 - I. TENNYSON'S POEMS. II. WORDSWORTH'S POEMS. III. POETRY AND CRITICISM. IV. ANGEL IN THE HOUSE. V. CARLYLE'S LIFE OF STERLING. VI. ESMOND. VII. MY NOVEL. VIII. BLEAK HOUSE. IX. WESTWARD HO ! X. WILSON'S NOCTES. XI. COMTE'S POSITIVE PHILOSOPHY. "One of the most delightful and precious volumes of criticism that has appeared in these days. ... To every cultivated reader they will disclose the wonderful clearness of perception, the delicacy of feeling, the pure taste, and the remarkably...
Página 37 - ... of Variations involving the variation of a certain double integral, the limits of the integration being also variable; it is the earliest example of the solution of such a problem. Gauss himself says on page 67, " Sed quum calculus variationum integralium duplicium pro casu ubi etiam limites tanquam variabiles spectari debent, hactenus parum excultus sit, hanc disquisitionem subtilem paullo profundius petere oportet.
Página 345 - Bjorling discusses another particular example before considering the general equation, namely, among all surfaces which can be formed by the motion of a straight line which always remains parallel to a fixed plane, to determine that of minimum area.
Página 286 - Let there be any linear differential expression which involves x, y, and the differential coefficients of y with respect to x...
Página 540 - We do not know a more readable book on a scientific subject, and it will be invaluable to young people, as well as interesting to those who are already acquainted with the subject it treats of.'— CLERICAL JOURNAL. LIFE AND CORRESPONDENCE OF M. DE TOCQUEVILLE. Translated from the French. By the Translator of 'Napoleon Correspondence with King Joseph.
Página 409 - ... that y is then a minimum and so is F. These results do not agree with those in the book. The case in which y = k seems there overlooked. If y = k we have h = k = a. And it may be seen that the relation on the 14th line of page 165 of the book may be satisfied by supposing a = y and the angle CPY zero. 352. On page 365 the following problem is suggested ; to construct upon a given base a curve such that the superficial area of the surface generated by its revolution round AB may be given, and...

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