652 REPOKT— 1892. 



and 



'' = N^, ^1^)- 



From equations (12) and (13), G. W. Hill has, with four different values of E, 

 found 0.' and y explicitly in terms t, for the particular solution in each case which 

 gives the simplest 07-bit (relatively to the revolving plane XOY), of which the one 

 which presents the greatest deviation from the well-known ' variational ' oval of 

 the elementary lunar theory, is a symmetrical curve with two outwardly pro- 

 jecting cusps corresponding to the moon in quadratures. He supposed this to be 

 the most extreme deviation from the variational oval possible for an orbit 

 surrounding the earth. Poincar6, in his ' M(5thodes Nouvelles de la M6canique 

 Celeste,' 1892, p. 109, admiring justly the manner in which Hill has thus ' si 

 magistralement ' studied the subject of finite closed lunar orbits, points out that there 

 are solutions corresponding to looped orbits, transcending Hill's, wrongly supposed 

 extreme, cusped orbit. Mr. Hill tells me that he accepts this criticism. The 

 labour of working out a fairly accurate analytical solution for any of Poincar6's 

 looped orbits, by Hill's method, would probably be very great. I have therefore 

 thought it might interest others besides ourselves to apply my graphic method to 

 the drawing of at least one of Poincar^'s looped orbits, in our Physical (and 

 Arithmetical) Laboratory in the University of Glasgow. Fig. 3 represents a 

 looped orbit, which has been worked out accordingly by Mr. Magnus Maclean, 

 Chief Official Assistant of the Professor of Natural Philosophy from the equations 

 (14), (15) above. 



8. B eduction of every prohlem of Two Freedoms in Conservative Dynamics, to 

 the drawing of Geodetic Lines on a Surface of given Specific Curvature. 

 By Lord Kelvin, Pres.B.S. 



1. Any conservative case of two-freedom motion is proved to be reducible to a 

 corresponding case of the motion of a material point in a plane. 



2. In plane conservative dynamics, with any given value for the energy constant, 

 E, the resultant velocity, q, at any point (.r, y) is a known function of Qi; y), being 

 given by the equation 



y^ = 2(E-V) .... (1) 



where V denotes the potential at (.r, y) ; and every problem depends on drawing 

 lines for which qds (the Maupertuis 'action') is a minimum. 



3. Considering any part, S, of the infinite plane, find a surface, S', such that 

 any infinitesimal triangle A' B' C drawn on it has its sides equal to 2 of those of 



a corresponding triangle A B C in the field, S, of our plane problem ; g^ denotmg 

 the value of y at any particular point (.i\„ «/„) in the plane. By the principle of 

 least action we see instantly that the lines on S', corresponding to paths on S, are 

 geodetic. Thus the adynatmc case of motion of a particle on S' is found as a 

 perfect and complete representative of the motion on the plane surface S, under 

 force with any arbitrarily given function, V, for its potential, and any particular 

 given value, E, for the total energy of the moving particle. 



4. It is proved easily that the surface, S', to be found according to §3, exists ; 

 and that its specific curvature (Gauss's name for the product of its two principal 

 curvatures) at any point is equal to 



goV log q 

 q' ' 



5. Examples are given of the finding of S'. As one example, illustrating the 

 practical usefulness of this method in dynamics, the problem of the parabolic 



