## The Quarterly Journal of Pure and Applied Mathematics, Volume 42J.W. Parker, 1911 - Mathematics |

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### Common terms and phrases

B₁ beix Bernoullian function Bernoullian numbers bounded variation C₁ Cesaro-Riesz coefficients Comptes Rendus condition conic conjugate constant content zero convergent convergent series corresponding cosh Crelle's Journ cross-ratio cubic curve denote derived determinant Determinante differential double points equations finite number formulę G. H. Hardy given Hence infinite integral J. W. L. GLAISHER L. E. DICKSON Math memoir motion MUIR mutually inscribed n-body problem obtained orbits pair paper Poincaré positive increment positive negative positive variation problem proved quadric quantities quartic relations involving repeated limit scroll set of content set of intervals set of points straight line summable surface tangent tetrads tetrahedra theorem theory three bodies three-body problem transformation U-formulę uneven united planes V₁ values variables x₁ Y₁

### Popular passages

Page 250 - Any two particles of matter in the universe attract one another with a force which is proportional to the product of the masses of the particles, and inversely proportional to the square of the distance between them.

Page 249 - Generalizations of the problem of several bodies, its inversion , and an introductory account of recent progress in its solution.

Page 294 - Bertrand, nnd he shows that every force under which a point describes a conic section, whatever be the initial conditions, either always passes through a fixed point or remains parallel to a fixed direction; finally, by applying his method to the determination of the explicit forms of the forces in the case of conic sections, Stephanos rediscovers the known results due to Darboux and Halphen.

Page 284 - ... the forces are derived from a potential function which may be written in the form1 1 On employing the usual substitutions the form given follows immediately from Oppenheim's solution in rectangular coordinates. See his memoir in the third volume of the Publications...

Page 208 - ... by the stipulation that the variable is not to cross the real axis at any point on the positive side of the origin.

Page 353 - ... Phys. (3), x, pp. 50-55.] This, being written by Kostka merely to compare his methods with those of Saalschutz as expounded shortly before in a 30-page paper,* naturally contains nothing fresh on our subject. The superiority which he claims is attributed to the use of determinants. MUIR, T.

Page 159 - For m = 3, w>3, .Fin a vanishing fonn.f in the present paper the field is that of integers taken modulo 2. In § 2 a theorem is obtained which enables us to write down at once every non-vanishing form. The problem of the separation of these non-vanishing forms into nonequivalent classes is treated in § 3— 8 for m = 4, n = 3 ; while for w = 6, n = 3, certain interesting classes are fully determined.

Page 84 - That this is not the case is shown by the following example which is due to Phillip Griffith.

Page 69 - ... to an upper semi-integral. For an upper semi-integral is lower semi-continuous on the right, and upper semi-continuous on the left. Cor. — If a success-ion of integrals of functions which are bounded below in their ensemble oscillates boundedh/, there is in every sub-succession a sequence of the integrals converging to an upper semi-integral.

Page 288 - Ql = -3 - (« which is thus made to appear as the unique case of conic section orbits for all initial conditions under forces varying as the masses and a function of the mutual distances. It may be observed here parenthetically that if a similar study be made for the cubic a first condition will be found to demand that the orbits be defined by equations of the form (13) aixt* + zbixfyi — 3aixtyl" — bt y(s — d = o, (1=1,2,3); the remaining analysis of the problem offers no difficulty.