Prof. H. Hennessy on Konayne's Cubes. 187 



Wlien^=0 ? 8111^=1/2-1, this, substituted in the value 



f ° l * d6 2 ' snows tnat tne latter must De negative ; hence 



sin 0=^2 — 1 gives for t a maximum, or = 24° 28' would 

 give the greatest thickness for t. Between this and zero the 

 thickness would give a smaller inclination and also a different 

 value for x. Jn the model x has been chosen between the 



two extreme values — ^ and |, or x= j (<s/2 + l). 



This value of x would allow greater values of t and 6 than 

 in the model, but they have been both determined by making 

 the greatest thickness of the flange at its extreme end equal 



to ,- . This gives 



2 ^2 ta ,n^ P 



2xs/2 



4:X 



with the assumed value of x, 



tan# = -—= • 



\/2 + l 



Hence 0=9° 4 5' nearly. 



The value of x numerically is a . * 6035 5 nearly. 



;(^2-l) 2 __a (3-2^2) 



p = a. 29289, t=° 



8x 



4(^2 + 1) 4 (1+^/2) ' 



The side of the cube is very nearly 1*92 inch, and the above 

 results agree with the measured values of x, p, 6, and very 

 nearly with that of t. 



It thus follows that a material solid cube can be so con- 

 structed as to allow of a cube of the same dimensions passing 

 through it by an aperture cut in the former without separa- 

 ting the remaining portions. As crystals are known to be 

 penetrated by others of similar shape, this problem may 

 possibly illustrate questions connected with the study of 

 isomorphous groups of the cubical type which are frequently 

 known to present the appearance of interpenetration. 



