LAWS OF DECREMENT. 309 



REGULAR TETRAHEDRON, 



Simple and mixed decrements on the angles. 



To determine the law of a simple or mixed decre- 

 ment on an angle of the regular tetrahedron, we may 

 assume as the unit of comparison, the ratio to an 

 edge a c, of a perpendicular a b upon the base, fig. 332, 

 drawn from the angle . The line a b measuring the 

 decrement in breadth, and the edge a c measuring 

 the decrement in height. 



The ratio of ab : ac is known, from the relation 

 of the tatrahedron to the cube, to be as *3 : 2. 



Let/g-, fig. 332, represent a secondary plane be- 

 longing to class &, whose inclination to the primary 

 plane, which is obviously equal to the angle b d e, 

 has been determined by measurement; we may call 

 this angle f 2 . 



In the triangle a d e, we have the following angles, 

 \/ ade=(lSO /,), 

 V d a e zz (90 |I t ) nr 54 44' 8", which we have 



called A,. 



V aed (/ a Aj. 

 Whence 

 ad : ae :: sin. (7 2 A z ) : sin. (180 7 2 ) :: pm : qn. 



In the fraction !?, - represents . 

 qn' n 2 



Dividing therefore sin * f-[J^_^) by _JL we shall 

 sin. 180 /.). g' 



find the values of p and gs jp, representing the decre- 

 ment in breadth on the angle a, proceeding along the 

 plane a b, and q the number of molecules in height 

 corresponding to the line a e. 



If the inclination on P, of any plane belonging to 

 class c, which rests on the edge between P' and the 



