38 On the Drawing of Figures of Crystals. 
13. Tetraxvonal system.—This system of crystallization includes 
those forms which contain three equal horizontal axes, at right an- 
gles with the vertical. The normal position of the horizontal axes 
is represented in fig. 6. The eye, placed in the line of the axis 
YY, observes two of the semiaxes, 
MZ and MU, projected in the same 
straight line, while the third MY ap- 
pears a mere point. ‘To give the 
axes a more eligible position for a rep- 
resentation of the various planes on 
a tetraxonal solid, we revolve them 
from right to left through a certain 
number of degrees, 5, and elevate 
the eye at an angle e«. The dotted 
lines in the figure represent the axes 
in their new situation, resulting from 
a revolution through a number of degrees equal tod = YMY’, In 
this position the axis MY’ is projected upon MP, MU’ upon MN 
and MZ’ on MH. Designating the intermediate axis I, that to the 
right IT, that to the left ILI, if the revolution is such as to give the 
projections of I and II, the ratio of 1:2, the relations of the three 
projections will be as follows: 1: IL: Il=1:2: 
he projection of 
I=sin 0d. 
Il=sin (60°—6), for MU N= U MY= YMU~— UMU'=60° —6. 
l11=sin (60° +3). 
From these equations and the above ratios, it follows that, 
sin (60° 4-5) =sin (60° —5)+sin 4; 
sin (60° —9)=2 sind; 
sin (60°-+6)=3 sin 6. 
ar] since (Trig. Anal.) 
in (60°-+46)= sin 60° cosd+sind cos 60°=2 sind; 
sin (60° —5)= sin 60° cos 5—sin 5 cos 60° =3 sin 5. 
Adding the equations we obtain 
2 sin 60° cosd=5 sind; 
2sin 60° 2/2 V3 
Whence tan d= 5 gt et 
cotd=5/1. 1 
From this equates we may deduce 
sind =f, ; 
-*. F=19° 6 24”, 
