THE PROBLEM OF THREE BODIES 



By F. W. HENKEL, B.A., F.R.A.S. 



The complete solution of the " problem of two spherical masses 

 moving under their mutual gravitation, their relative positions 

 and motions being known, at any one instant to determine their 

 motions at any other time," was obtained by Newton more than 

 two hundred years ago and given to the world in the first book 

 of his immortal Principia. He showed that the bodies would 

 move in similar paths round their common centre of gravity 

 (or mass) and these paths would be one or other of the curves 

 known as the conic sections, ellipse (circle as special case), 

 parabola or hyperbola, the speed at any point determining 

 which of these curves would be described. When one body is 

 greatly more massive than the other it is often found convenient 

 to imagine it reduced to rest and refer the motion of the other 

 body to it, rather than to the common centre of gravity, by 

 applying to both bodies a motion equal and opposite to the 

 actual motion of the heavier one. 



If now instead of two bodies only we have three or more 

 whose positions and masses are known, the general problem of 

 determining their future relative positions far transcends the 

 present power of our mathematical methods and seems likely 

 to long remain insoluble. However, for all the cases we have 

 to deal with in our system, where we have two bodies moving 

 round their common centre of gravity and the third body either 

 very much smaller than the central one or very distant, approxi- 

 mate methods are available whereby a solution to any required 

 degree of accuracy may be obtained. We have in the Planetary 

 theory a predominant central body, the Sun (whose mass exceeds 

 that of all the planets together, more than seven hundred times), 

 round which revolve much smaller bodies, the planets. The 

 motion of each of these latter will be mainly determined by the 

 action between itself and the sun, so that to a first approxima- 

 tion we may consider the planet as moving in an ellipse round 

 its central body. Then by the principle of the "superposition 



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