The Scientific Papers of Sir George Darwin: Periodic Orbits by George Howard Darwin

By George Howard Darwin

Sir George Darwin (1845-1912) was once the second one son and 5th baby of Charles Darwin. After learning arithmetic at Cambridge he learn for the Bar, yet quickly back to technological know-how and to Cambridge, the place in 1883 he was once appointed Plumian Professor of Astronomy and Experimental Philosophy. His relations house is now the site of Darwin university. His paintings was once involved basically with the impact of the solar and moon on tidal forces in the world, and with the theoretical cosmogony which developed from useful statement: he formulated the fission idea of the formation of the moon (that the moon used to be shaped from still-molten subject pulled clear of the Earth via sun tides). This quantity of his gathered papers covers periodic orbits and a few miscellaneous papers, together with investigating the healthiness records of the wedding of first cousins - of curiosity to a member of a dynasty during which such marriages have been universal.

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32) + 2 - i f H J + n + 8p . 3H . 3f ^ dpds 3s2 These are the equations of motion in the varied orbit. The variation of the last of (30), the Jacobian integral, gives V8V=8p~ + 8sd^ (33) dp ds Now 8 V is the tangential velocity of the point a: + 8%, y + 8y in the varied orbit, relatively to the original point x, y. But as we only want to consider a velocity relatively to the axes of * and y, which themselves rotate with angular velocity n, our p, s axes must be regarded as rotating with angular velocity d/dt, instead of (d/dt) + n.

I shall accordingly use the value of c which lies nearest to \/<&0 as fundamental. We have j u s t arrived at a physical meaning for c by considering the principal term in the series; now in so doing we were in effect considering only the mean motion of the body with reference to the moving axes; therefore \e is also the ratio of the synodic to the anomalistic period*. If T denotes the synodic period, the mean motion of the body referred to axes fixed in space is (2TT/T) + n; and if dwjdt denotes the mean angular velocity of the pericentre with reference to axes fixed in space, the mean motion of the body with reference to the pericentre is (2TT/T) +n — (dw/dt).

3 . , - 3s m S cos 0 + —s sin i|r cos i r p 3 . V2I2 = 2 (1; + 1 ) + -, + -. Hence ' and T6 p6 ^— = cos2 0 ^-— + 2 sin d> cos (/> ^ - 5 - + sin2 cf> Therefore V —=- = ^— = — sin d> -=—V cos Also since Vds~ Y -t) (40) This completes the formula for M> in terms of the coordinates, the velocity, the curvature and of <$>. It may be useful to obtain the expressions for Bs and B in terms of the new independent variable s. D iv. 3 34 PERIODIC ORBITS. [1, § 10 The second of (37) may be written down at once, namely (41) Also it is clear from geometrical considerations that s whence , d8p S = - * |™Z - ^ § 10.

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