Transactions of the Cambridge Philosophical SocietyUniversity Press, 1912 |
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الصفحة vii
... equal to the sum of the limits of the terms of the series ? ( ii ) when is the integral of the sum of an infinite series equal to the sum of the integrals of the terms ? Or in symbols , ( i ) when is ( ii ) when is ∞ lim Σ fn ( x ) = Σ ...
... equal to the sum of the limits of the terms of the series ? ( ii ) when is the integral of the sum of an infinite series equal to the sum of the integrals of the terms ? Or in symbols , ( i ) when is ( ii ) when is ∞ lim Σ fn ( x ) = Σ ...
الصفحة 8
... equal to Σ ( − 1 ) a „ & * ( 9 ) = & „ G [ * 2 * $ ( 2 ) da ; lim Σ ( -1Ý * ( 7 ) , lim 20 , [ " 70 Σαπ € ~ * 2 5 * $ ( 1 ) dx . us the question ( 6 ) may also be answered in the affirmative . 3. Thus we arrive at : THEOREM II . We may ...
... equal to Σ ( − 1 ) a „ & * ( 9 ) = & „ G [ * 2 * $ ( 2 ) da ; lim Σ ( -1Ý * ( 7 ) , lim 20 , [ " 70 Σαπ € ~ * 2 5 * $ ( 1 ) dx . us the question ( 6 ) may also be answered in the affirmative . 3. Thus we arrive at : THEOREM II . We may ...
الصفحة 32
... equal to ** sin ( 2n + 1 ) x 0 sin a f ( sin x ) dx . In particular , if 0 < a < 2 , sin ax x ( iv ) Suppose ( as in § 20 ) that and consider the integral ᅲ ƒ ( sin3 æ ) dæ = \ π [ " ′′ ƒ ( sin3 a ) dæ . П Xo ( X , T ) = e− ( 7 + mi ) ...
... equal to ** sin ( 2n + 1 ) x 0 sin a f ( sin x ) dx . In particular , if 0 < a < 2 , sin ax x ( iv ) Suppose ( as in § 20 ) that and consider the integral ᅲ ƒ ( sin3 æ ) dæ = \ π [ " ′′ ƒ ( sin3 a ) dæ . П Xo ( X , T ) = e− ( 7 + mi ) ...
الصفحة 47
... equal to ( 72 + a2 + B2 ) 1 - B a a - arc tan T2 + ( a + B ) * + arc tan T T d ( ( 72 + a2 + a2 ) $ ) da ( 72 + ( a + a ) 2 ) da . It is easy to see that the last integral is continuous for 70 , so that the expression tends , as 70 , to ...
... equal to ( 72 + a2 + B2 ) 1 - B a a - arc tan T2 + ( a + B ) * + arc tan T T d ( ( 72 + a2 + a2 ) $ ) da ( 72 + ( a + a ) 2 ) da . It is easy to see that the last integral is continuous for 70 , so that the expression tends , as 70 , to ...
الصفحة 49
... equal length , this should be simply the time required by a wave to travel twice the length of either rod . The experiments of Voigt § and others had , however , shewn that this is influenced by the nature of the ends of the rods where ...
... equal length , this should be simply the time required by a wave to travel twice the length of either rod . The experiments of Voigt § and others had , however , shewn that this is influenced by the nature of the ends of the rods where ...
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عبارات ومصطلحات مألوفة
a₁ absolutely convergent applied asymptotic B₁ bounded function bracket C₂ CAMBRIDGE concomitant consider continuous function convergent corresponding cos² curve denote differential coefficient differential equation dx² elastic equal expression factor finite follows formula ƒ x Hence impact infinite integrand irreducible k₁ Lebesgue integral Lemma linear monotone monotone function obtain operator oscillation P₁ plastic positive pressure prove R₁ reciprocant repeated integrals respect result s₁ satisfied sequence shew solution summable suppose symbols theorem theory transformation U₂ uniformly uniformly convergent unique limit upper V₁ values variables velocity W. H. Young w₁ zero α₁ ах дг ди др ду дх მყ
مقاطع مشهورة
الصفحة 280 - We know, however, that this integral is also equal to the sum of the residues at the poles of the integrand /
الصفحة 286 - The sum of the residues at the poles of the last factor, which occurs in (13.18) is therefore equal to minus the sum of the residues at the poles of the other factors.
الصفحة 358 - Since the modulus of a sum is not greater than the sum of the moduli...
الصفحة 96 - In support of such an hypothesis we may remark that as the crystalline grains are built up from their various centres, there will in general be a certain number of molecules to spare between their surfaces ; and these molecules will evidently be distributed in the spaces between the grains with relatively small density, ie, in such a fluid or serai-fluid condition, as would just account for the viscous reaction required.
الصفحة 140 - ... the same properties as the forms upon which they effectively operate, and that the forms which are invariant and the operators are properly to be regarded as one system of forms possessing the same properties.
الصفحة 312 - A geometrical progression is a series in which the ratio of any term to the preceding term is the same for all terms of the series.
الصفحة 126 - Plemelj's canonical form for the principal part of the solving function of an integral equation of the second kind in the neighbour hood of a pole.
الصفحة 48 - would prove independent of the length of the rods, provided the velocity and other conditions of the impact were kept unchanged. This was found to be the case, so that, by plotting duration of impact against length of rods, a straight line was obtained, whose slope gave the required value of the wave-velocity. * " On the Effects of Momentary Stresses in Metals,
الصفحة 348 - Rules for determining when the process of reversing the order of integration in a repeated integral is allowable.
الصفحة 100 - We are concerned here with the variation of integral forms, or more specially with their invariance, under the action of an operator which is an extension of the hydrodynamical operator in Enler's equations.