444 
Or, expressing ju and v in terms of the polar distances C 2 ofl~ 
-PzPiVs and G 2°A= —PaPiPv 9 
And cos ju^cos 3 ^— cos 2 j9 2 + cos 2 ^> 3 , 
C 2 d 5 o 1 = P2P1P3 OAPi^PaPtPv 
cos v = cos p 2 cos £> 3 + cos 'P\'P' 2 i + cos lhlh > 
formulae calculable at once by Byrne^s dual logarithims, or 
easily adapted to logarithmic computation by subsidiary angles. 
All the formulae for the pentagonal dodecahedrons are 
immediately derivable from those of the irregular twenty -four - 
faced trapezohedron. 
144. Limits of the Form of the Irregular Twenty -four-faced 
Trap ezohedron . 
As m and n approach in magnitude to unity, the irregular 
twenty-four-faced trapezohedron approximates to the octa- 
hedron ; and when m and n both equal unity, it becomes the 
octahedron. In this case the three planes meeting in the points 
Op o 2 , &c., o g (fig. 25, Plate III.), lie in the same plane, and 
the edges, such as G^ v C 2 S V lie in the same line. 
As m and n both increase in magnitude and become infinitely 
great, this form approximates to and becomes the cube. In 
this case the four planes meeting in G v C 2 , &c., 0 6 , become 
the same plane, and the edges, such as o 4 Vp °A° 5 > tlie 
same straight line. t . 
As m approaches to unity while n increases m magnitude 
and becomes infinitely great, the form approaches the rhombic 
dodecahedron. When m equals unity, while n remains finite, 
the form becomes the three-faced octahedron. Allien m and n 
equal each other and are both finite and greater than unity, 
the form becomes that of the regular twenty-four-faced 
trapezohedron. Finally, when m remains finite and greater 
than unity and n becomes infinite, the form becomes that of 
the pentagonal dodecahedron. 
145. As yet the half-symmetrical forms with parallel faces, 
the pentagonal dodecahedron and the irregular twenty-four- 
faced trapezohedron have only been found in combination with 
those of the full symmetrical forms of the cubical system, and 
never with those of the half- symmetrical forms with inclined 
faces. 
146. For the pentagonal dodecahedrons the following are 
the values of the angles /t and v. 
E i ["I m= 77° 19' v=60° 48'. 
P | njoc,] m =73°44' v=61° 19'. 
G4 [If 00 ] ^=67° 23' v=62° 31'. 
H | [1 loo] M =53° 8' v=66° 25'. 
m| [13 00 ] m =36°52' v=72° 83'. 
N f [1 4 00 ] ju=28° 4' v= 76° 23'. 
