ON THE TABLES OP MODIFICATIONS. 99 



fications of the cube, may be conceived to be com- 

 prised within two natural limits ; the one being the 

 primary plane of the cube, and the other the plane a. 

 For it is obvious that the inclination of b on P may 

 approach nearly to 180 , but it can never reach that 

 limit. And it may pass from such an obtuse angle, 

 through an almost unlimited series of planes inclining 

 less and less on P, until at length the inclination 

 would be nearly the same as that of a on P ; but it is 

 apparent that it can never attain this limit, there 

 being only the one plane , which can incline to the 

 primary planes at that angle. 



The series of planes belonging to class c of the 

 cube, are limited within the planes a and e. The 

 angle at which the planes marked c incline to each 

 other, may be conceived to increase, until it ap- 

 proaches very near to 180; the three planes would 

 then very nearly become one, and similar to , but 

 they can never reach that limit. And they may also 

 be conceived to incline more and more upon the edge, 

 until at length they would very nearly coincide with 

 the plane e, but they could never exactly coincide 

 with that plane. 



This may be readily comprehended if we refer to 

 the tables, and remark that each of the series of new 

 solids which would result from the series of planes 

 belonging to class c, would be contained within 24 

 planes, while only one single solid contained within 

 8 planes could be produced from class , and only 

 one other by class , contained within 12 planes; 

 neither of which therefore could fall within the series 

 resulting from class c. 



The series of individual modifications belonging 

 to class d of the cube, may be conceived to lie within 

 several limits, according to the direction in which the 

 change of position of the modifying planes may be 



