## Analytical View of Sir Isaac Newton's Principia |

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angle applied apsides attraction axis Bernouilli body moves calculus central force centre of forces centre of gravity centrifugal force centripetal force circle co-ordinates comet conic sections corollaries cycloid demonstration density described diameter differential differential calculus direction discoveries distance disturbing force earth eccentricity ellipse equal equation evanescent expression fluid fluxion focus force acting force is inversely formula geometrical given points Hence hyperbola infinitely integral investigation Laplace law of attraction length mass maxima and minima method moon moon's move round observed orbit oscillation osculating circle parabola particle pendulum perpendicular planets Principia problem proportion proposition quadrature quantity radii radius of curvature radius vector ratio rectangle resistance respecting revolve round satellites Scholium Sir Isaac Newton solution space sphere square straight line supposed surface syzygy tangent theory tide tion trajectory triangles varies velocity wave whole

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Page 61 - ... the squares of the periodic times are as the cubes of the distances from the common centre, the centripetal forces will be inversely as the squares of the distances.

Page 323 - ... reason to believe resists the motion of comets; loaded, perhaps, with the actual materials of the tails of millions of those bodies, of which they have been stripped in their successive perihelion passages, and which may be slowly subsiding into the sun.3* In 1849, Herschel added several significant remarks to this account.

Page 358 - In this case, small displacements of this knife-edge will not materially alter the position of the centre of gravity or radius of gyration of the pendulum about an axis through its centre of gravity. The time of swing about the fixed knife-edge will therefore remain practically constant. The best determination of the correct position of the movable knife-edge for an equal time of oscillation will be given when for...

Page 378 - We thus obtain the force of gravity at the level of the sea, supposing all the land above this level were cut off and the sea constrained to keep its present level. As the...

Page 42 - Therefore the force F is directly as the versed sine, and inversely as the square of the time. From this it follows that the central force may be measured in several ways. The arc being QC, we are to measure the central force in its middle point P. Then the areas being as the times ; twice the triangle SPQ...

Page 259 - For now the particles 990 of the water do not all of them pass through the hole perpendicularly, but, flowing down on all parts from the sides of the vessel, and converging towards the hole, pass through it with oblique motions; and in tending downwards meet in a stream whose diameter is a little smaller below the hole than at the hole itself...

Page 82 - Forty-first Proposition when he found those methods unmanageable. This would naturally confirm him in his plan of preferring geometrical methods ; though it is to be observed that this investigation, as well as the inverse problem for the case of rectilinear motion in the preceding section, is conducted more analytically than the greater part of the Principia, the reasoning of the demonstration conducting to the solution and not following it synthetically. A is the height from which a body must fall...

Page 335 - ... of the earth's radius ; in which case this nutation might become much greater than for the solid spheroid. 4. In addition to the above motions of precession and nutation, the pole of the earth would have a small circular motion, depending entirely on the internal fluidity. The radius of the circle thus described would be...

Page 278 - From whence it is probable that the lengths of the pulses in all sounds made in open pipes are equal to twice the length of the pipes.

Page 346 - It becomes then of the utmost importance to inquire in what cases this supposition may be made. Now Lagrange enunciated two theorems, by virtue of which, supposing them true, the supposition may be made in a great number of important cases, in fact, in nearly all those cases which it is most interesting to investigate.