BY MARTIN GAKDIXER, C.E. 



2 ' ( 



From these we at once obtain the relation 



(d, M 2 (d ,h \ 2 (c b\* (c , b ,Y 



\ 1 I/ V +l +!/ \ 1 I/ \ +l n+iy 



And now since the square of a chord of a sphere is equal to 

 the product of the diameter and the perpendicular let fall from 

 one extremity of the chord on the tangent plane at the other 

 extremity, we perceive (from the last formula) that the following 

 relation (adopting the notation employed in theorem 5) subsists, 



rf l A l d n+V A +l C I> A l C n+l> A + l 



viz. 



NOTE. 



Sir William Hamilton, the Astronomer Bo} T al of Ireland, has 

 given much attention to the problem of this paper. He pub- 

 lished the results of his researches in the Philosophical Magazine 

 for July 1849, and afterwards drew the attention of the Mathe- 

 matical Section of the British Association to the subject. He 

 succeeded in solving only the particular case in which the sur- 

 face is an ellipsoid and the closed n'gon even sided. 



It seems that his " Quaternion " and other symbolical methods 

 led him to infer that independent of the two positions for the 

 first angular points of the closed w'gons, which may be real or 

 imaginary according to peculiar states of the data, there are also 



I 



