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.— Trazté des Fonctions Ellip- 
tiques, tom. iii. page 97. It may be seen that they follow at 
once from these expansions, if we remember that 
a log (1 + 2acosx + a”) dx =0 
0 
when a is less than unity; a theorem proved by Poisson in 
the seventeenth cahier of the Journal de [ Ecole Polytech- 
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 
