inscribed in a Sphere. 383 



of opposite planes separately. A cone can in general be drawn 

 through the two circles of intersection, and the truncated por- 

 tion of any four-sided pyramid inscribed in this cone will give 

 an inscribed hexahedron in the sphere. Moreover, if a qua- 

 drangle be inscribed in one of the circles, as ABCD, and a plane 

 be drawn through AB cutting the other circle in ab, and then two 

 more planes be drawn through the points «AD and bBC cutting 

 the second circle in cd respectively, it will be seen that the qua- 

 drangle CBcd will not be plane unless — 



(a) The plane AB«6, and hence the three others, pass through 

 the vertex of the cone, or 



08) Either pair of lines AB and CD or AC and BD be par- 

 allel to one another and perpendicular to the common 

 diametral plane of the cone and sphere. 



(a) is the general case, and the hexahedron is a frustum of a 

 quadrangular pyramid. By considering in like manner the 

 other two pairs of planes, it will be seen that the hexahedron is 

 the common frustum of three four-sided pyramids. 



(J3) is the singular case of a four-sided prism, the two sections 

 being equally inclined, but in opposite directions. Note : that 

 this prism will not in general be inscribed in a right cylinder. 



2. In the general case it obviously follows that the hexahedrou 

 has six diagonal planes, passing two by two through the three 

 vertices of the pyramids. 



3. The four diagonal lines of the hexahedron intersect in a 

 point. This follows, in the general case, from their lying two 

 by two on the six diagonal planes, and it is easily seen in the 

 singular case. 



4. Hence in both cases there are six diagonal planes, all inter- 

 secting in a point. 



5. This point is the spherical pole to the plane passing through 

 the three vertices. In the singular case the polar plane passes 

 through the parallel lines of intersection. This property may- 

 be deduced from the similar one of plane quadrilaterals inscribed 

 in a circle. 



6. Since the sections of the cone by the planes ABCD, abed 

 are subcontrary, the corresponding angles ka, Bb, Cc, &c. are 

 equal each to each. Hence the sum of the opposite angles 

 BAD and bed is equal to two right angles, and so of similarly 

 opposite pairs. This is not, in general, true of the singular case. 



7. Since six planes intersect two by two in fifteen lines, every 

 hexahedron must have, associated with it, three external lines of 

 intersection. 



(a) If these three associated lines lie in one plane, they must 

 intersect in three points, and then the hexahedron will have six 

 diagonal planes and four diagonal lines all intersecting in one 



