322 APPENDIX -CALCULATION OF THE 



RHOMBIC DODECAHEDRON. 



Intermediary decrements on the obtuse solid angles. 



On referring to class h of the modifications of this 

 form, we shall perceive that two planes rest on the 

 edge between P and P". 



Let the plane which inclines most on P be mea- 

 sured on P and on P". Call its inclination on P, / 3 , 



onP"./ 4 . 



The symbol representing this plane would be 



(Bp B'q B"r). 



To determine /?, <?, and r, we must, as we have 

 done for the tetrahedron, find the two plane, angles 

 of the defect, which we shall call A . and A 6 , by the 

 formulae 



sin. i A 5 = 



Y sin. (180 J 3 )sin. /, 



sin. A fi = 



V 



-CQ8.K (180*- J 4 )+ J +(180"- 1 3 ) ]co8.ft[ (180- J 4 )+ 1 , -(180- J 3 ) ] 



J 4 )sin. /, 



Whence^ : q :: sin. [180 (A v -\-A 5 )~\ : s\n.A 5 

 p : r :: sin. [180_(y/ 1 -j-^ 6 )] : sin.^ 6 

 And the particular values of p, q, and r thus found, 

 being substituted for those letters in the above gene- 

 ral symbol, we shall obtain the particular symbol of 

 the observed plane. 



It thus appears that the formulae used for deter- 

 mining the laws of decrement on the obtuse solid 

 angles of this figure, are similar in character to those 

 applied to the determination of decrements on the 

 solid angles of the tetrahedron. 



This results from an analogy which subsists between 

 these solid angles in the two forms. For the lines 



