309 ° 
vary inversely as the cube of the perpendicular and linear dis- 
tance from a fixed plane passing through the centre. 
The first theorem had been suggested to Sir W. Hamilton 
by a recently resumed study of a part of Sir Isaac Newton’s 
Principia; and he had been encouraged to seek for the second 
theorem, by a recollection of a result respecting motion in a 
spherical conic, which was stated some years ago to the Aca- 
demy by the Rev. C. Graves. In that result of Mr. Graves, 
the fixed pole was a focus of the conic, and the polar was 
therefore the director arc; consequently, the sine of the dis- 
tance from the polar was proportional to the sine of the distance 
from the pole, and, instead of the law now mentioned to the 
Academy, there was the simpler law of proportionality to the 
inverse square of the sine of the distance from the fixed pole 
or focus. 
Professor Graves observed, that he had that morning, in 
conversation with the President, stated the theorem just an- 
nounced, respecting the motion of a material point on the sur- 
face of a sphere. Sir William Hamilton having, at the last 
meeting of the Academy, kindly communicated to him his 
theorem of plane central forces, it occurred to Professor 
Graves to inquire whether two theorems, which he had stated 
in January, 1842,* relating to the motion of a point in a sphe- 
rical conic, might not be included in a more general one, ana- 
logous to that first mentioned by Sir William Hamilton. This 
inquiry led him to perceive the truth of Sir William Hamil- 
ton’s second theorem. 
The mode of proof employed by Professor Graves rests, 
so far as regards the dynamical part of the question, on the 
two following elementary propositions : 
If a material point, P, constrained to move on the surface 
of a sphere, be urged by a force acting along a great circle 
passing through a fixed point, s; 
* See Proceedings of the Academy, vol. ii. p. 209. 
