Mr. C. M. Willich on the Partition of the Cube. 27 



have been supposed, especially if the following precautions be 

 taken. Let the distance of the disk B from the focus o be so 

 measured that the diverging cone of rays exactly fills up the pass- 

 ing holes. In this case there is in this mode of observation also a 

 maximum of brightness of momentary duration for each hole. The 

 explanations given above on the distinctness of the optical images 

 apply also in this case. If instead of a cone of rays a fine cy- 

 linder of rays be used which falls upon suitably combined tuning- 

 forks with steel mirrors, Lissajou's luminous figures may be ob- 

 jectively represented, and resolved as above into one or more 

 moving points of light. That, further, in these experiments sun- 

 light within certain limits may be replaced by artificial light is a 

 matter of course. 



IV. On the Partition of the Cube, and some of the Combinations 

 of its parts. By Charles M. Willich, late Actuary and Se- 

 cretary to the University Life Assurance Society*. 



A CUBE may be divided into equal and uniform bodies in 

 various ways. 



1st. By lines from the centre to the eight angles of the cube, 

 which will give six four-sided pyramids (B). 



2nd. By lines from one of the upper angles of the cube drawn 

 diagonally to the opposite angles, dividing the cube into three 

 equal and uniform solids. Each of these solids being halved, 

 forms a left- and a right-handed solid. The six bodies thus 

 produced, though equal in mass, differ in shape so far, that three 

 may be termed left-handed and three right-handed, in the same 

 way as the hands of the human body. 



3rd. By lines drawn from the centre to four angles of the cube, 

 and continued on each face, will be produced four equal and similar 

 bodies (G), each composed of two three-sided pyramids united at 

 their base — the one having the same angle as the trihedral roof 

 of the Bee's cell, viz. 109° 28' 16", the other 90°. These four 

 bodies rearranged produce the half of a dodecahedron with rhom- 

 boidal faces. 



4th. Another division of the cube may be made producing 

 the tetrahedron and octahedron-, viz., diagonal lines from two 

 of the upper angles of the cube, continued on the other faces, 

 will cut off four three-sided pyramids, leaving in the centre a 

 tetrahedron. The four three-sided pyramids cut off may be so 

 arranged as to produce the half of the true octahedron. 



The four-sided pyramid obtained by the first mode of division 



* Communicated by the Author, having been laid before the British 

 Association, Nottingham, August 1866. 



