332 Mr Levy on the Modes of Notation 
two faces, such as abd , b dc, meeting in an edge, in the same di- 
rection as one of the oblique diagonals of the rhomboid, will be 
represented by (i:i) ; that of two faces, such as hdc y dee , meet- 
ing in an edge situated in the same direction as one of the supe- 
rior edges of the rhomboid, will be represented by (i . i ) ; and, 
finally, (i , i) will designate the incidence of one of the faces,, 
such as abd , upon the corresponding face abd' of the inferior 
pyramid. It is easy to demonstrate, that, in every dodecaedron 
derived from a rhomboid, there exists between these three angles 
the very simple relation expressed by the equation, 
sin \ (i , i) = cos \ (i : i) + cos | (i . i) 
By means of which, two of these incidences being known, the 
third will be immediately found, especially as the value of any 
one of these three, deduced from the above equation, may, with- 
out difficulty, be transformed into another, to which logarithmic 
calculation may be applied. 
Now, to resolve the proposed problem. The values of the 
angles ( i : i ), ( i . £), (i , i), should be expressed in terms of x, 
y , #, or rather the values of these last quantities in terms of the 
first. But the calculations necessary to be gone through to obtain 
them are very long ; and the formulas themselves are, besides, 
so complicated, as to be of very little use. Their comparison 
leads, however, to a simple result, which is sufficient to re- 
solve most of the questions referring to dodecaedrons derived 
from a rhomboid, and which I shall demonstrate in a direct 
manner, without using the above mentioned formulae. 
Draw the oblique diagonals r o, r p, Fig. 7. and let them 
meet fh , gh in l and i. Join fi , gl meeting in &, and draw 
the axis rki J of the rhomboid. It is obvious that the angle 
of the two planes fglhfk r' is equal to ( i . i), and the angle 
of the two planes f g h , Ik r' is equal to J ( i : i)'; moreover 
the angle of the two planes fk Ikr' is equal to 60°. We 
shall have, therefore, by spherical trigonometry, in the triangu- 
lar solid angle whose summit is at k , and formed by the three 
planes f k l or fg h,fk r', Ik r\ the two following equations : 
cos i (i . i) . sin fk l = cos Ik / . sin f k / — J sin l k r\ cos fkr r 
cos | (i : i) . sin fk l = cos fk r ' . sin Ik r' — | sin fk r'. cos lkr\ 
