13 $ 
of Weiss , Mohs, and Haiiy. 
&nd dividing the first by the second, 
cos \ (i . i) __ £ tang fk r' — tang Ikr 
cos i (i :i) 2 tang l k r' — tan gfk r' 
We shall obtain, consequently, the value of the ratio of these 
two cosines, if we can get those of the tangents of the angles fkr’ 
and Ikr'. It is even sufficient to determine the value of 
tan gfk r', for, in changing in it oc into 2 , and 2 Into sc, we shall 
get the tangent of gkr and by changing the sign of this, the 
tangent of Ik r'. 
From /and m, let fq, m s, be drawn perpendicular upon rr f , 
let rs — a, and ms — p , then rq = - , fq — 2 , 
SC * X 
rk =z 
S a 
x +y 4 - 
consequently k q = 
a{y±z- 2x) 
(x+y + z)x’ and 
tangfk y = 
P oc+y + z 
a ‘ y 4 - z — %x’ 
and tang7 k r f — 
P 
a 
a? + y + z 
y +x — 2z' 
These values being substituted in the expression gives, 
cos j (i . i) y — z 
cos ^ (i : i) ~~ x — y° 
This formula will give at once a simple relation between the 
three unknown quantities x, y, z, when the two angles (i . i) 9 
(i : i) are known. It is also a test of the simplicity of the in- 
dices of the secondary planes, which we are’ considering ; for if 
these indices, that is x, y, z, are always simple numbers, it ne- 
cessarily follows that or its equal, the ratio of the co~ 
x z 
sines of half the two pyramidal angles of any dodecahedron de- 
rived from any rhomboid, is always a simple integral, or frac- 
tional number ; a result the correctness of which I have had fre- 
quent opportunities to verify. 
It is now easy to apply the preceding formula to the dode- 
caedrons which result from simple decrements, by assuming pro- 
per values for x, y, and z. Thus, by taking x — o, y = 1 and 
z — n, the formula will correspond to the case of a dodecaedron 
produced by n rows in breadth on the superior edges of the 
