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 

 Princijjia ; 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. 



