. 134". OF THE CONNEXION OF FORMS. 



This method of resolution produces the same result as if 

 the first and the second, or the first and the third, had been 

 applied to the tetracontaoctahedron. The first requires 

 the faces of the subordinate points to disappear ; the others 

 require in this case only the enlargement of the alternating 

 faces of the remainder. 



If in the icositetrahedron considered above, we enlarge 

 those faces which have disappeared, and vice versa, let those 

 disappear which have been enlarged before ; the result 

 in respect to that obtained first, will be a Left tetrahedral 

 pentagonal-dodecahedron. But the icositetrahedron may be 

 resolved both in the normal and in the inverse position. 

 Hence both the differences, as to Right and Left, and as 

 to Normal and Inverse, come into consideration in the te- 

 trahedral pentagonal-dodecahedron. 



The trigrammic tetragonal-icositetrahedron may be re- 

 solved after the first process, by enlarging all the faces 

 contiguous to its principal points, &c. Each of these faces; 

 is intersected by five others, two of which belong to the same, 

 the other three to adjacent principal points. For the rest, 

 every thing is as above ; and the trigrammic tetragonal- 

 icositetrahedron yields exactly the same fourths. 



The pentagonal-icositetrahedron is resolved according to 

 the first method, by enlarging all the faces contiguous to 

 the principal points, &c. Each of these faces again is in- 

 tersected by five others, and the result of the resolution is 

 likewise a tetrahedral pentagonal-dodecahedron. 



Thess four pentagonal-dodecahedrons, different on one 

 side as to right and left, on the other as to their normal 

 or inverse position, reproduce in binary combinations the 

 icositetrahedron s, and in a quadruple combination the tetra- 

 contaoctahedron itself, from the resolution of whicli they 

 have been obtained. 



The first of these differences is expressed by the letters 

 r and 1, the second by the signs -4- and , prefixed to 



the general notation of one-fourth of the tetraooiHa- 

 4 



octahedron. The four dodecahedrons will therefore be ; 



