PHILOSOPHICAL SOCIETY OF WASHINGTON. 107 



This paper is expected to appear in full in the Astronomische 

 Nachrichten. 



Remarks upon this paper were made by Messrs. E. B. Elliott 

 qnd W. B. Taylor. 



Mr. Marcus Baker then presented the following communication 



ON A GEOMETRICAL QUESTION RELATING TO SPHERES. 



On January 17, 1882, Mr. Doolittle called the attention of the 

 Society to the geometrical problem To determine a circle equally dis- 

 tant from four given points in a plane, and showed that the state- 

 ment in Chauvenet's Geometry, (p. 308, Ex. 110,) that this problem 

 admits of four solutions is erroneous, there being in general fourteen, 

 solutions. The extension of this problem to spheres and five points 

 in space is nearly as simple as for the case of circles and four 

 points in a plane. 



Let it be proposed to solve the following : 



Problem. — To determine a sphere equally distant from five given 

 points. 



The distance to a sphere, considered here, is to be measured along 

 a diameter, produced if necessary, and hence for any position we 



have two distances, one a maximum, the other a minimum. 



• 



Solution. — Case I. Through any four of five given points, a, b, c, 

 d, e, as, for example, b, c, d, e, describe a sphere ; the fifth point, a, 

 will in general fall within or without this sphere, of which call the 

 radius R and centre C ; also, let c( be the distance from the centre 

 of this sphere to the point a. Then two spheres described with 

 centre C and radii $(Tl ± o() fulfil the condition of being equidis- 

 tant from the five points. 



Every distinct group of four of the five given points in like 

 manner gives two solutions ; hence of this kind there are in all ten 

 solutions. 



Case. II. Through any three of the five given points, a, b, c, d, e, 

 as a, b, c, pass the circumference of a circle; from the centre of the 

 circle erect a perpendicular. This perpendicular is the locus of all 

 points equidistant from points a, b, c. Join the points d and e by a 

 line ; bisect this line by a plane perpendicular thereto. This plane 

 is the locus of all points equidistant from d and e. The intersec- 

 tion of these two loci is the centre of two spheres equidistant from 

 the five points. 



