September. 1910. 



KNOWLEDGE. 



351 



solution. Dr. Hill, in ISZf). pointed out a new- 

 method of treating the problems of the Lunar motion, 

 and he took as the first approximation, not the 

 ellipse, but what he called the variational curve as 

 the Moon's orbit. The Moon is considered to move 



Figure 2. Equipotential surfaces about a double star with 

 equal components of equal mass (after Darwin). 



in a circle distorted by the Sun's attraction, and this 

 orbit is called a " periodic one," its period being one 

 sj-nodic month {i.e.. inter\-al from New Moon to 

 New Moon or I'^ill Moon to Full Moon, 29i davs). 

 The eccentric form and the change of inclination 

 (the Moon's path is not in the plane of the Earth's 

 orbit but is inclined at an angle of about 5" to it) 

 are "free" oscillations about the periodic orbit, the 

 " annual equation " (arising from the varN-ing distance 

 of the Sun at different times of the \-ear) is a 

 " forced " oscillation. M. Poincare, M. G_vlden, and 

 Dr. Charlier have continued this subject from the 

 purely mathematical point of view, whilst Sir George 

 Darwin has also dealt with it numericalh' in a 

 comparatively simple manner, and we propose to give 

 a brief account of some of his work. He begins b\- 

 considering the case of three bodies, called respectively 

 the Sun, a planet moving round it which he calls 

 Jove, and a third body whose mass is infinitesimal 

 (so that it will produce no appreciable effect upon 

 the motions of the other two) which he calls the 

 planet, or satellite. Jove (J), of unit mass, moves 

 round the Sun. of mass* ten, in a circle, and all three 

 bodies lie in one and the same plane. He first 

 obtains what is called the equation of relative energ\- 

 (the well-known Jacobian integral V^^2f2 — C). 

 Since for real motion \"^ must be positive, 20 must 

 be greater than, or at least equal to, C, and so the 

 planet can never cross the curve represented b\' 

 2f2 = C, which agrees with Dr. Hill's result in 

 assigning definite limits to the past and future position 

 of the Moon in our s\stem. He then considers the 

 form of the curx'es obtained hv giving different 



\, lines to the constant C, called the constant of relative 

 energy. For large values of C they are closed ovals 

 round S and J respectively with an outer branch 

 round both S and J. The larger oval, shrinking as C 

 becomes less, unites with the inner oval round J, and 

 the curve becomes of horse-shoe shape. The horse- 

 shoe then narrows at the middle and breaks into two 

 elongated portions, these gradually shrinking into 

 two points equidistant from S and J. Figure 1 shows 

 the form of the curves for the critical values.' 



It was shown by Lagrange 

 \'ears ago, that there is an exact 



more than a hundred 

 solution of the 

 problem of three bodies when they are each at one 

 angle of an equilateral triangle which revolves 

 uniformly, and since two triangles of equal altitude 

 may be drawn on the same base (one above and the 

 other below S J ) , there are two such positions. This is 

 approximately realised in our system. .\ minor planet 

 recently discovered revolves at about the distance of 

 Jupiter from the Sun, and this arrangement appears to 

 be a stable one. A paper on this relation appeared 



A. Orbit of Satellite leaving Jove and passing luider the 

 control of the Sun. 



(Thick line, orl)il of Satellite. Path of Jove, dotted line). 



B. Referred to moving axes. 

 Figure 3. (After Darwinl. 



some time since in " Knowledge." In addition to 

 the above determinations, Sir George Darwin has also 

 considered the case of a satellite leaving Jove 

 and passing under the control of the Sun, and 

 Figure 3 shows the path for C=o9, referred both 

 to axes fixed in space and to revolving axes. The 

 full curve shows the path of the planet, the dotted 



'" The value of ten for tlie mass of the Sun in terms of Jove is taken for the purpose of exaggerating 

 of perturbation as compared with those in our system, so as to give a clearer view of their nature. 



t \\'c may regard these curves as the traces on the plane of the paper of surfaces of zero velocity, 

 bodies in one plane, the problem is reduced to a two dimensional one. 



all the phenomena 

 By taking all three 



