308 



In conclusion it may be observed, that the particular re- 

 sults, (4), (6), (7), (8), are nothing more than immediate con- 

 sequences of Mr. Jacobi's factorial developments of the trigo- 

 nometrical functions of the amplitude of an elliptic function, 

 in terms of the function itself. — Traite des Fonctions Ellip-^ 

 tiques, torn. iii. page 97. It may be seen that they follow at 

 once from these expansions, if we remember that 



riog(i 



2acos X + a^) dx — 



when a is less than unity ; a theorem proved by Poisson in 

 the seventeenth cahier of the Journal de I'Ecole Poly tech- 

 nique. 



Sir William R. Hamilton stated the following theorems of 

 central forces, which he had proved by his calculus of quater- 

 nions, but which, as he remarked, might be also deduced from 

 principles more elementary. 



If a body be attracted to a fixed point, with a force which 

 varies directly as the distance from that point, and inversely 

 as the cube of the distance from a fixed plane, the body will 

 describe a conic section, of which the plane intersects the fixed 

 plane in a straight line, which is the polar of the fixed point 

 with respect to the conic section. 



And in like manner, if a material point be obliged to re- 

 main upon the surface of a given sphere, and be acted on by 

 a force, of which the tangential component is constantly di- 

 rected (along the surface) towards a fixed point or pole upon 

 that surface, and varies directly as the sine of the arcual dis- 

 tance from that pole, and inversely as the cube of the sine of 

 the arcual distance from a fixed great circle ; then the material 

 point will describe a spherical conic, with respect to which the 

 fixed great circle will be the polar of the fixed point. 



Thus, a spherical conic would be described by a heavy 

 point upon a sphere, if the vertical accelerating force were to 



