TRANSACTIONS OF SECTION A. 



649 



sponding to the position of Qi- With this radius, find the position of the centre 

 of curvature, C„ in PiC„L, the line of the radius through P. With this centre 

 of curvature, and the fresh radius of curvature, describe an arc PiPjQ., making 

 P Q equal to about half the length intended for the third arc ; calculate radius 

 of curvature for Q.,; draw an arc P3P3Q3; and contmue the procedure. This 

 process is well adapted for finding orbits by the ' trial and error ' method described 

 in my article 'On Some "Test Cases" of the Maxwell-Boltzmann Doctrine 

 reo-arding Distribution of Energy,' sect. 13 ; 'Proc. Royal Soc.', June 11, 1891. 



°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 



dt- 



-yx- 



1=- 



df^ 



xij- 



Fm. 2. 



The initial point Pq 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 y 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 y 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 gravity. 

 Let (x,y) be co-ordinates of the moon relative to OX in line with the sun, out- 

 wards, and OY perpendicular to it in the direction of the earth's orbital motion. 

 The weU-toiown equation of motion relatively to revolving co-ordinates gives, for 



