of Dynamical Problems. A 1 5 



The accompanying curve (fig. 2) has been drawn with 

 great care, and with very interesting success, in the ' trial 

 and error' method of finding the first and simplest orbit, by 

 my secretary, Mr. Thomas Carver, for the case of motion 

 defined by the equations 



d 2 x 2 



Fiff. 2. 



The initial point P was taken on one of the lines cutting the 

 axes of x and y at 45°, and at first at a random distance from 

 the origin. A trial curve was worked according to the 

 method described above, and was found to cut the axis of x 

 at an oblique angle. Other trial curves, with unchanged 

 energy-constant, were worked from initial points at greater 

 or less distances from the origin, until a curve was found to 

 cut the axis of x perpendicularly. This curve is one-eighth 

 part of the orbit ; and is shown in fig. 2 repeated eight 

 times in order to complete the orbit, which is symmetrical on 

 the two sides of the axes of x and y. 



As an interesting case of motion related to the Lunar 

 Theory, suppose the mass of the moon be infinitely small in 

 comparison with the mass of the earth ; and the earth and sun 

 to have uniform motions in circles round their centre of 



