A Commentary on Newton's Principia: With a Supplementary Volume. Designed for the Use of Students at the Universities, Volume 2Black, Young and Young, 1833 - 415 pages |
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arbitrary constants axes axis become bodies center of gravity coefficients comet consequently consider contain coordinates cosines curve determine differential equations distance dx² elements ellipse equal equation of partial equilibrium errors expression fixed plane fluid formula function given gives Hence homogeneous function integral latitude longitude manner mass mean motion meridian multiplied node observations obtain orbit parabola parallel partial differences perihelion perpendicular point attracted positive preceding equation PROB quantities radius radius-vector relatively right ascension shell sines and cosines sphere spheroid squares straight line substitute suppose surface tion variable variations whence
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Page 178 - ... that the mean longitude of the first satellite, minus three times that of the second, plus twice that of the third, is always equal to two right angles.
Page 334 - Required a proof. 10. Compare the space described in one second by the force of gravity in any given latitude, with that which would be described in the same time, if the earth did not revolve round its axis. 11. At the extremities of the arms of a given lever are suspended respectively the weights p and q; supposing p to descend, and the lever to revolve round the fulcrum ; it is required to determine the angular velocity of the lever at any point of its descent from an horizontal to a vertical...
Page 50 - ... negative, there is one negative root and no positive root; and hence we conclude that equation (55) can have but one real root. We may dispense with the direct solution of this equation by forming a table of the values of v corresponding to those of t — T in a parabola whose perihelion distance is equal to the mean distance of the earth from the sun. This table will give the time corresponding to the anomaly v in any parabola, whose perihelion distance is q, by multiplying by q*, the time which...
Page 254 - These phenomena are the variation of the degrees of the meridian, and of gravity, the precession of the equinoxes, the nutation of the terrestrial axis, the inequalities which the flattening...
Page 334 - ... space in the least time. 16. A body having descended from a given altitude, begins to descend along the arc of a given cycloid whose base coincides with an horizontal plane ; find the points where it will leave the curve, and where it will meet the horizontal plane. 17. Find the ratio of the times of oscillation of a pendulum, at the equator, and at the pole, supposing the Earth to be a sphere, and to revolve about its axis in a given time. 18. If the first fluxions of a common ordinate of a...
Page 357 - ... 8. Require the nature of the curve such, that if any point P in it be taken, and an ordinate PN and normal PG be drawn to the axis; then if the triangle PNG be placed in such a position that the sub-normal NG may become the ordinate, PG will be the normal.
Page 332 - ... convex surface, determine the focus of parallel rays after refraction at each lens. 13. The excentricity of the earth's orbit being small, the variation of the angular velocity is nearly proportional to the cosine of the angle made by the radius vector and perihelion distance. Required a proof. 14. From the poles to the equator, the decrease of the length of a pendulum always vibrating in the same time, varies as the square ,of the cosine of latitude. Required proof. 15. If the altitude of a...
Page 1 - Hence, in the first place, if the bodies which compose a system move uniformly in right lines, then their common center of gravity will either remain at rest, or will move uniformly in a right line ; for if a body equal to the sum of the bodies were placed in that center, and then acted upon by the same forces which cause the bodies to move separately in right lines, it would either remain at rest, (viz. when the forces counteract each other,) or would describe uniformly the remaining...
Page 358 - Required the latitude of the place. 3. If any number of circles be drawn through two given points A and B cutting a given circle, the lines which join the points of intersection shall all meet AB produced in the same point. 4.
Page 52 - What we need for the tug of war value (f,/f.)> is the ratio of the mass of the planet to that of the sun...